Regressão Linear

### Modelos estatísticos - Aula de modelos lineares
### Pacotes usados na aula.
require("ISLR")
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require("ggplot2")
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require("GGally")
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require("leaps") ## seleção de variaveis
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require("car")
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require(tidyverse)
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Recomendações de leitura

[…]

Objetivo

Num modelo de regressão linear, a relação entre uma variável aleatória de interesse (variável resposta) e um conjunto de preditores (variáveis explicativas) é definida por uma função linear envolvendo os preditores e um conjunto de parâmetros associados (coeficientes ou $$′s ou parametros).

“Composição”

  • Variaveis Resposta
  • Variaveis Explicativas

Exemplo:

  • Aplicação 1: Preço de Imóveis
    • Variável Resposta: Preço do imóvel
    • Variáveis Explicativas: Tamanho do imóvel (m2), número de quartos, localização (bairro), idade do imóvel, presença de garagem
  • Aplicação 2: Desempenho Escolar
    • Variável Resposta: Nota final em uma disciplina
    • Variáveis Explicativas: Número de horas estudadas por semana, frequência nas aulas, nota em provas anteriores, participação em atividades extraclasse, apoio familar
  • Aplicação 3: Vendas de um produto
    • Variável Resposta: Vendas mensais de um produto
    • Variáveis Explicativas: Preço do produto, gastos em marketing, número de vendedores, promoções realizadas, condições econômicas.
  • Aplicação 4: Consumo de Energia
    • Variável Resposta: Consumo de energia elétrica (kWh)
    • Variáveis Explicativas: Tamanho da residência (m2), número de moradores, uso de eletrodomésticos, temperatura média na região, tipo de aquecimento
  • Aplicação 5: Rendimento de Investimentos
    • Variável Resposta: Rendimento de uma carteira de investimentos
    • Variáveis Explicativas: Taxa de juros, porcentagem de ações vs. renda fixa, valor inicial investido, taxa de inflação, duração do investimento

Modelo de regressão linear

Seja \(y\) a variável resposta e \(x1, x2, ..., xk\) os \(k\) preditores. O modelo de regressão linear fica definido por: \[ y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_k x_k + \epsilon \]

onde:

  • \(\beta_0\) é o intercepto do modelo (se todos os \(x\)s e \(\epsilon\) forem 0 $ y = _0$ );
  • \(\beta_1, \beta_2, ... \beta_k\) são parâmetroa do modelo, que correspondem aos efeitos lineares de cada um dos \(k\) preditores;
  • \(\epsilon\) representa os erros aleatórios, específicos para cada indivíduo. É a parte de \(y\) que não é explicada pela regressão linear.

Erros

Embora sejam não observáveis e não explicados pelo modelo, na regressão linear assumimos as seguintes propriedades para os erros:

  • Os erros têm média zero;
  • Os erros têm variância constante;
  • Os erros para dois indivíduos quaisquer são não correlacionados;
  • Os erros têm distribuição normal.

Este conjunto de suposições é usualmente denotado por \(\epsilon \sim N (0, \sigma^2)\).

Regressão Linear Simples

Como caso particular da regressão linear temos a regressão linear simples, que se caracteriza por considerar um único preditor:

\[ y = \beta_0 + \beta_1 x + \epsilon, \space \space \space \space \space \epsilon \sim N (0, \sigma^2) \]

Regressão Linear Polinomial

Outro modelo de regressão linear bastante usual é o modelo de regressão polinomial, que permite explicar uma relação não linear entre a resposta e o(s) preditor(es):

\[ y = \beta_0 + \beta_1 x + \beta_2 x^2 + ... + \beta_k x^k + \epsilon, \space \space \space \space \space \epsilon \sim N (0, \sigma^2) \] Fig 01

Fig 02
Fig 02
Fig 03
Fig 03
Fig 04
Fig 04
Fig 05
Fig 05
Fig 06
Fig 06

Um breve histórico

  • Século 19: Desenvolvimento da teoria de mínimos quadrados, que serve de base para o ajuste de modelos de regressão.
  • A teoria de mínimos quadrados teve origem na Física motivada, dentre outros, por problemas na área navegação (século 18).
  • Ao longo do século 19, modelos lineares e o método de mínimos quadrados passaram a ser utilizados em outras ciências, baseados em modelos pré-estabelecidos, ou apenas em evidências empíricas.
  • O termo regressão foi introduzido por Francis Galton em 1875, baseado no princípio da “regressão à média”.

Regressão linear

  • Os dois objetivos principais de uma análise de regressão são os seguintes:
    • Exploratório- Analisar a relação funcional entre uma variável aleatória (resposta) e um conjunto de variáveis preditoras.
    • Preditivo- Predizer valores não observados da variável resposta para valores especificados das variáveis preditoras.

Exemplo- Consumo de combustível

  • Neste exemplo vamos analisar os dados da base Auto, disponíveis na biblioteca ISLR do R.
  • A base de dados dispõe de informações técnicas de 392 modelos de automóveis das décadas de 1970 e 1980, como consumo de combustível. potência do motor, dimensões dentre outras.
  • O objetivo aqui é ajustar modelos de regressão linear que expliquem o consumo de combustível com base em características do modelo.
  • Neste primeiro momento, para fins ilustrativos vamos considerar apenas duas variáveis de cada vez, embora o mais usual seja considerar múltiplas variáveis conjuntamente numa análise de regressão.
summary(Auto)
##       mpg          cylinders      displacement     horsepower        weight    
##  Min.   : 9.00   Min.   :3.000   Min.   : 68.0   Min.   : 46.0   Min.   :1613  
##  1st Qu.:17.00   1st Qu.:4.000   1st Qu.:105.0   1st Qu.: 75.0   1st Qu.:2225  
##  Median :22.75   Median :4.000   Median :151.0   Median : 93.5   Median :2804  
##  Mean   :23.45   Mean   :5.472   Mean   :194.4   Mean   :104.5   Mean   :2978  
##  3rd Qu.:29.00   3rd Qu.:8.000   3rd Qu.:275.8   3rd Qu.:126.0   3rd Qu.:3615  
##  Max.   :46.60   Max.   :8.000   Max.   :455.0   Max.   :230.0   Max.   :5140  
##                                                                                
##   acceleration        year           origin                      name    
##  Min.   : 8.00   Min.   :70.00   Min.   :1.000   amc matador       :  5  
##  1st Qu.:13.78   1st Qu.:73.00   1st Qu.:1.000   ford pinto        :  5  
##  Median :15.50   Median :76.00   Median :1.000   toyota corolla    :  5  
##  Mean   :15.54   Mean   :75.98   Mean   :1.577   amc gremlin       :  4  
##  3rd Qu.:17.02   3rd Qu.:79.00   3rd Qu.:2.000   amc hornet        :  4  
##  Max.   :24.80   Max.   :82.00   Max.   :3.000   chevrolet chevette:  4  
##                                                  (Other)           :365
ggplot(Auto, aes(x = year)) + geom_histogram()
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

ggplot(Auto, aes(x = year, y = mpg)) + geom_point() +
    stat_smooth(method = "lm") +
    theme_bw(base_size = 14)
## `geom_smooth()` using formula = 'y ~ x'

plot1 <- ggplot(Auto, aes(x = horsepower, y = mpg)) + geom_point() + theme_bw(base_size = 14)
plot1

plot1 + coord_trans(x="log2", y="log2")

Fig 07
Fig 07
  • Inicialmente, vamos ajustar um modelo de regressão linear que permita explicar o consumo de combustível (mpg- variável resposta) em função do ano de lançamento do modelo (yearvariável explicativa).

  • A Figura 7 sugere relação linear crescente entre o consumo de combustível e o ano de lançamento do modelo.

  • O modelo de regressão linear para esse par de variáveis fica especificado por: \[ mpg = \beta_0 + \beta_1 \times year + \epsilon, \] onde \(\beta_0\) e \(\beta_1\) são os parâmetros do modelo (intercepto e inclinação da reta de regressão) e \(\epsilon\) representa os erros aleatórios.

  • O ajuste da regressão linear consiste na estimação dos parâmetros do modelo (β0 e β1), com base nos dados amostrais, que produzem a reta de regressão que melhor se ajusta aos dados.

  • O método usual de estimação dos parâmetros de uma regressão linear é o método de mínimos quadrados, que será estudado adiante.

  • Aplicando o método de mínimos quadrados, obtemos os parâmetros estimados \(\hat{\beta_0} = −70.01\) e \(\hat{\beta_1} = 1.23\), produzindo a seguinte reta de regressão ajustada: \[ \hat{mpg} = −70.01 + 1.23 \times year \]

Fig 08
Fig 08
Fig 09
Fig 09

[continua….]

Exemplo prático- Taxa de formados

  • Na parte prática desta aula vamos analisar dados sobre a taxa de formados em \(n=777\) universidades norte-americanas no ano de 1995.
  • O objetivo é ajustar um modelo de regressão linear que permita explicar a taxa de alunos formados em função das características das universidades e de seus alunos.
  • Uma breve descrição das variáveis explicativas é apresentada na sequência.

Variaveis

  • Grad.Rate: Taxa de alunos formados (resposta);
  • Apps: Número de alunos inscritos;
  • Accept: Número de alunos aceitos;
  • Enroll: Número de novos alunos matriculados;
  • Top10perc: Percentual de novos estudantes entre os 10% melhores no ensino médio;
  • Top25perc: Percentual de novos estudantes entre os 25% melhores no ensino médio;
  • F.Undergrad: Número de alunos em período integral nos cursos de graduação;
  • P.Undergrad: Número de alunos em período parcial nos cursos de graduação;
  • Outstate: Número de alunos bolsistas de outros estados;
  • Room.Board: Gastos de hospedagem e alimentação;
  • Books: Gastos com materiais bibliográficos;
  • Personal: Gastos com recursos humanos;
  • PhD: Percentual de professores com doutorado;
  • Terminal: Percentual de professores com grau terminal;
  • S.F.Ratio: Razão alunos/professor;
  • perc.alumni: Percentual de ex-alunos que contribuem com donativos;
  • Expend: Gasto educacional por aluno.
  • Privada ou publica

CODIGO

### Carregamento e visualização inicial da base

data("College") ### Carregando a base
#help("College") ### Acessando a documentação

head(College,10) ### Visualizando as dez primeiras linhas
##                              Private Apps Accept Enroll Top10perc Top25perc
## Abilene Christian University     Yes 1660   1232    721        23        52
## Adelphi University               Yes 2186   1924    512        16        29
## Adrian College                   Yes 1428   1097    336        22        50
## Agnes Scott College              Yes  417    349    137        60        89
## Alaska Pacific University        Yes  193    146     55        16        44
## Albertson College                Yes  587    479    158        38        62
## Albertus Magnus College          Yes  353    340    103        17        45
## Albion College                   Yes 1899   1720    489        37        68
## Albright College                 Yes 1038    839    227        30        63
## Alderson-Broaddus College        Yes  582    498    172        21        44
##                              F.Undergrad P.Undergrad Outstate Room.Board Books
## Abilene Christian University        2885         537     7440       3300   450
## Adelphi University                  2683        1227    12280       6450   750
## Adrian College                      1036          99    11250       3750   400
## Agnes Scott College                  510          63    12960       5450   450
## Alaska Pacific University            249         869     7560       4120   800
## Albertson College                    678          41    13500       3335   500
## Albertus Magnus College              416         230    13290       5720   500
## Albion College                      1594          32    13868       4826   450
## Albright College                     973         306    15595       4400   300
## Alderson-Broaddus College            799          78    10468       3380   660
##                              Personal PhD Terminal S.F.Ratio perc.alumni Expend
## Abilene Christian University     2200  70       78      18.1          12   7041
## Adelphi University               1500  29       30      12.2          16  10527
## Adrian College                   1165  53       66      12.9          30   8735
## Agnes Scott College               875  92       97       7.7          37  19016
## Alaska Pacific University        1500  76       72      11.9           2  10922
## Albertson College                 675  67       73       9.4          11   9727
## Albertus Magnus College          1500  90       93      11.5          26   8861
## Albion College                    850  89      100      13.7          37  11487
## Albright College                  500  79       84      11.3          23  11644
## Alderson-Broaddus College        1800  40       41      11.5          15   8991
##                              Grad.Rate
## Abilene Christian University        60
## Adelphi University                  56
## Adrian College                      54
## Agnes Scott College                 59
## Alaska Pacific University           15
## Albertson College                   55
## Albertus Magnus College             63
## Albion College                      73
## Albright College                    80
## Alderson-Broaddus College           52
dim(College) ### Acessando a dimensão da base
## [1] 777  18
summary(College) ### Resumo das variáveis
##  Private        Apps           Accept          Enroll       Top10perc    
##  No :212   Min.   :   81   Min.   :   72   Min.   :  35   Min.   : 1.00  
##  Yes:565   1st Qu.:  776   1st Qu.:  604   1st Qu.: 242   1st Qu.:15.00  
##            Median : 1558   Median : 1110   Median : 434   Median :23.00  
##            Mean   : 3002   Mean   : 2019   Mean   : 780   Mean   :27.56  
##            3rd Qu.: 3624   3rd Qu.: 2424   3rd Qu.: 902   3rd Qu.:35.00  
##            Max.   :48094   Max.   :26330   Max.   :6392   Max.   :96.00  
##    Top25perc      F.Undergrad     P.Undergrad         Outstate    
##  Min.   :  9.0   Min.   :  139   Min.   :    1.0   Min.   : 2340  
##  1st Qu.: 41.0   1st Qu.:  992   1st Qu.:   95.0   1st Qu.: 7320  
##  Median : 54.0   Median : 1707   Median :  353.0   Median : 9990  
##  Mean   : 55.8   Mean   : 3700   Mean   :  855.3   Mean   :10441  
##  3rd Qu.: 69.0   3rd Qu.: 4005   3rd Qu.:  967.0   3rd Qu.:12925  
##  Max.   :100.0   Max.   :31643   Max.   :21836.0   Max.   :21700  
##    Room.Board       Books           Personal         PhD        
##  Min.   :1780   Min.   :  96.0   Min.   : 250   Min.   :  8.00  
##  1st Qu.:3597   1st Qu.: 470.0   1st Qu.: 850   1st Qu.: 62.00  
##  Median :4200   Median : 500.0   Median :1200   Median : 75.00  
##  Mean   :4358   Mean   : 549.4   Mean   :1341   Mean   : 72.66  
##  3rd Qu.:5050   3rd Qu.: 600.0   3rd Qu.:1700   3rd Qu.: 85.00  
##  Max.   :8124   Max.   :2340.0   Max.   :6800   Max.   :103.00  
##     Terminal       S.F.Ratio      perc.alumni        Expend     
##  Min.   : 24.0   Min.   : 2.50   Min.   : 0.00   Min.   : 3186  
##  1st Qu.: 71.0   1st Qu.:11.50   1st Qu.:13.00   1st Qu.: 6751  
##  Median : 82.0   Median :13.60   Median :21.00   Median : 8377  
##  Mean   : 79.7   Mean   :14.09   Mean   :22.74   Mean   : 9660  
##  3rd Qu.: 92.0   3rd Qu.:16.50   3rd Qu.:31.00   3rd Qu.:10830  
##  Max.   :100.0   Max.   :39.80   Max.   :64.00   Max.   :56233  
##    Grad.Rate     
##  Min.   : 10.00  
##  1st Qu.: 53.00  
##  Median : 65.00  
##  Mean   : 65.46  
##  3rd Qu.: 78.00  
##  Max.   :118.00
### Vamos considerar Grad.Rate (taxa de formados) como a variável resposta na nossa análise. Começamos a análise com alguns gráficos.

ggplot(College, aes(x = Grad.Rate)) + geom_histogram() +
    theme_bw(base_size = 14)
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

### Distribuição das taxas de formados

ggplot(College, aes(x = Top10perc, y = Grad.Rate)) + geom_point() +
    geom_smooth(method = "loess") + 
    theme_bw(base_size = 14)
## `geom_smooth()` using formula = 'y ~ x'

### Taxas de formados versus percentual de alunos entre os 10% melhores
### no ensino médio.

ggplot(College, aes(x = Outstate, y = Grad.Rate)) + geom_point() +
    geom_smooth(method = "loess") + 
    theme_bw(base_size = 14)
## `geom_smooth()` using formula = 'y ~ x'

### Taxas de formados versus investimentos externos.

ggplot(College, aes(x = perc.alumni, y = Grad.Rate)) + geom_point() +
    geom_smooth(method = "loess") +
    theme_bw(base_size = 14)
## `geom_smooth()` using formula = 'y ~ x'

### Taxas de formados versus porcentagens de ex-alunos contribuintes.
ggpairs(College, proportions = "auto") ### Matriz de gráficos de dispersão. - DEMORA PARA RODAR

#para salvar o ultimo grafico
ggsave("plot_grande.png", device = "png", width = 100, height = 80, units = "cm")
# tamanho maximo 50 in ou 126 cm
Grafico pesado
Grafico pesado
ggcorr(College[,-1], label = TRUE, label_round = 2) ### Correlograma.
ggsave("plot_colorido.png", device = "png", width = 30, height = 20, units = "cm")
Grafico colorido
Grafico colorido
### Parte 1 - Ajuste dos modelos lineares. Comecemos com o caso de apenas uma 
### variável explicativa (no caso, perc.alumni)

### Para ajustar modelos lineares no R usamos a função lm. Vamos consultar
### a documentação da função.

#help('lm')

Multipla (1 var resposta) e Multivarida (mais var respostas) são coisas diferentes

### Ajuste da regressão linear simples (assumindo relação linear entre a taxa de formados e o percentual de ex-alunos contribuintes)

ajuste1 <- lm(Grad.Rate ~ perc.alumni, data = College) ###(resposta ~ var.explicativas)

ajuste1
## 
## Call:
## lm(formula = Grad.Rate ~ perc.alumni, data = College)
## 
## Coefficients:
## (Intercept)  perc.alumni  
##     49.9863       0.6805

\[ \hat{Grad.Rate} = 49,9863 + 0,6805 \times perc.alumni \] chapeu indica estimado?

summary(ajuste1)
## 
## Call:
## lm(formula = Grad.Rate ~ perc.alumni, data = College)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -58.247  -9.513   0.043   9.362  54.404 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 49.98633    1.12345   44.49   <2e-16 ***
## perc.alumni  0.68049    0.04338   15.69   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.98 on 775 degrees of freedom
## Multiple R-squared:  0.241,  Adjusted R-squared:   0.24 
## F-statistic: 246.1 on 1 and 775 DF,  p-value: < 2.2e-16
### O percentual de ex-alunos contribuintes tem efeito positivo, e 
### estatisticamente significativo na taxa de formados.
### Vamos visualizar o ajuste do modelo
ggplot(College, aes(x = perc.alumni, y = Grad.Rate)) + geom_point() +
    stat_smooth(method = "lm") +
    theme_bw(base_size = 14)
## `geom_smooth()` using formula = 'y ~ x'

### Vamos investigar possível efeito quadrático do percentual de contribuíntes na taxa de formados. Para isso, adicionamos ao preditor o termo quadrático da variável explicativa, da seguinte forma:

ajuste2 <- lm(Grad.Rate ~ perc.alumni + I(perc.alumni^2), data = College)
summary(ajuste2)
## 
## Call:
## lm(formula = Grad.Rate ~ perc.alumni + I(perc.alumni^2), data = College)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -57.444  -9.146   0.042   9.448  53.446 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      45.896244   1.877789  24.442  < 2e-16 ***
## perc.alumni       1.085948   0.155614   6.978 6.42e-12 ***
## I(perc.alumni^2) -0.007652   0.002821  -2.712  0.00683 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.91 on 774 degrees of freedom
## Multiple R-squared:  0.2481, Adjusted R-squared:  0.2462 
## F-statistic: 127.7 on 2 and 774 DF,  p-value: < 2.2e-16
### O termo quadrático é estatisticamente significativo, indicando que a
### relação entre as variáveis não é linear. Vamos dar um passo além, e
### incluir o termo de terceira ordem para o percentual de contribuintes
### (modelo cúbico).

\[ \hat{Grad.Rate} = 45.896244 + 1.085948 \times perc.alumni -0.007652 \times perc.alumni^2 \]

de acordo com os valores de p é significativo, então manteria essa variavel

ajuste22 <- lm(formula = Grad.Rate ~ ., data = College)
summary(ajuste22)
## 
## Call:
## lm(formula = Grad.Rate ~ ., data = College)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -53.897  -7.132  -0.292   7.213  54.056 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 33.8736716  4.8480858   6.987 6.15e-12 ***
## PrivateYes   3.3813758  1.6965147   1.993 0.046605 *  
## Apps         0.0012984  0.0004418   2.939 0.003390 ** 
## Accept      -0.0006961  0.0008627  -0.807 0.419995    
## Enroll       0.0021593  0.0023081   0.936 0.349814    
## Top10perc    0.0548964  0.0717587   0.765 0.444501    
## Top25perc    0.1351288  0.0549667   2.458 0.014179 *  
## F.Undergrad -0.0004712  0.0004008  -1.176 0.240138    
## P.Undergrad -0.0014836  0.0003902  -3.802 0.000155 ***
## Outstate     0.0010174  0.0002334   4.359 1.49e-05 ***
## Room.Board   0.0019143  0.0005908   3.240 0.001246 ** 
## Books       -0.0022205  0.0029168  -0.761 0.446739    
## Personal    -0.0016635  0.0007698  -2.161 0.031000 *  
## PhD          0.0872827  0.0568102   1.536 0.124859    
## Terminal    -0.0747023  0.0623172  -1.199 0.231002    
## S.F.Ratio    0.0758222  0.1593102   0.476 0.634254    
## perc.alumni  0.2793343  0.0491750   5.680 1.91e-08 ***
## Expend      -0.0004565  0.0001542  -2.961 0.003163 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 12.75 on 759 degrees of freedom
## Multiple R-squared:  0.4615, Adjusted R-squared:  0.4495 
## F-statistic: 38.27 on 17 and 759 DF,  p-value: < 2.2e-16
## PrivateYes -> 1 se privada (YES); 0 se publica(No)

## Nesse modelo (ajustado com todas as outras variaveis) as escolas privadas tem p<0.05, portanto ser privada tem um impacto  no (maior) número de formados (3.3 pontos percentuais)

ajuste23 <- lm(formula = Grad.Rate ~ Private , data = College)
summary(ajuste23)
## 
## Call:
## lm(formula = Grad.Rate ~ Private, data = College)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -53.998 -10.042   0.002  11.002  49.002 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   56.042      1.112  50.406   <2e-16 ***
## PrivateYes    12.956      1.304   9.937   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 16.19 on 775 degrees of freedom
## Multiple R-squared:  0.113,  Adjusted R-squared:  0.1119 
## F-statistic: 98.74 on 1 and 775 DF,  p-value: < 2.2e-16
## Num modelo só com a privada ela (por "acaso") tambem teve p<0.05 mas com um impacto diferente (12.9 pontos percentuais)
confint(ajuste22) ### Intervalo de confiaça
##                     2.5 %        97.5 %
## (Intercept) 24.3564214869 43.3909217410
## PrivateYes   0.0509572771  6.7117943696
## Apps         0.0004312157  0.0021656167
## Accept      -0.0023897796  0.0009975317
## Enroll      -0.0023717122  0.0066902689
## Top10perc   -0.0859726333  0.1957655121
## Top25perc    0.0272239131  0.2430337318
## F.Undergrad -0.0012581212  0.0003156791
## P.Undergrad -0.0022496029 -0.0007176822
## Outstate     0.0005592115  0.0014756792
## Room.Board   0.0007545476  0.0030741345
## Books       -0.0079464883  0.0035055569
## Personal    -0.0031746541 -0.0001524155
## PhD         -0.0242410359  0.1988064637
## Terminal    -0.1970367830  0.0476322483
## S.F.Ratio   -0.2369186844  0.3885630948
## perc.alumni  0.1827990643  0.3758695198
## Expend      -0.0007590979 -0.0001538281

\[ \hat{Grad.Rate} = 33,87 + 3,38 \times (Private = Yes) + 0.0012984 \times (Apps) \]

Variavel Dummie: (relevel)

  • Sexo: F, M -> Sexo Feminino (1 se feminino, 0 se masculino)

  • Escolaridade: Sem_Escolaridade, EF, EM, ES ->

    • EF (1 ou 0)
    • EM (1 ou 0)
    • ES (1 ou 0)
    • Sendo 1 TEM e 0 Não tem, consequentemente os 3 zeros é Sem_Escolaridade

\[ \hat{y} = \hat{\beta_0} + \hat{\beta_1} \times EF + \hat{\beta_2} \times EM + \hat{\beta_3} \times ES \] - Sem escolaridade: \(\hat{y} = \hat{\beta_0}\)

  • Ens. Fund: \(\hat{y} = \hat{\beta_0} + \hat{\beta_1}\)

  • Ens. Med: \(\hat{y} = \hat{\beta_0} + \hat{\beta_2}\)

  • Ens. Sup: \(\hat{y} = \hat{\beta_0} + \hat{\beta_3}\)

Se for EM/EF: \[ \hat{y} = (\hat{\beta_0} + \hat{\beta_1}) - (\hat{\beta_0} + \hat{\beta_2}) = \hat{\beta_1} - \hat{\beta_2} \]

\[ y = \beta_0 + \beta_1 x + \varepsilon \] \[ \varepsilon = (y - (\beta_0 + \beta_1 x)) \] \[ residuo = y - (\hat{\beta_0} + \hat{ \beta_1 } x) \] Erro é o “real”, residuo é o estimado.

ajuste3 <- lm(Grad.Rate ~ perc.alumni + I(perc.alumni^2) + I(perc.alumni^3), data = College)
summary(ajuste3)
## 
## Call:
## lm(formula = Grad.Rate ~ perc.alumni + I(perc.alumni^2) + I(perc.alumni^3), 
##     data = College)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -56.571  -9.123   0.005   9.346  52.896 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      42.9519252  2.8823421  14.902  < 2e-16 ***
## perc.alumni       1.5681680  0.3905839   4.015 6.53e-05 ***
## I(perc.alumni^2) -0.0276219  0.0151030  -1.829   0.0678 .  
## I(perc.alumni^3)  0.0002297  0.0001706   1.346   0.1787    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.91 on 773 degrees of freedom
## Multiple R-squared:  0.2499, Adjusted R-squared:  0.247 
## F-statistic: 85.84 on 3 and 773 DF,  p-value: < 2.2e-16
### O termo de ordem cúbica não tem significância estatística. Vamos seguir
### a análise com o modelo quadrático.

\[ \hat{Grad.Rate} = 42.9519252 + 1.5681680 \times perc.alumni -0.0276219 \times perc.alumni^2 + 0.0002297 \times perc.alumni^3 \] de acordo com os valores de p é significativo, então manteria essa variavel

### Vamos extrair alguns elementos do modelo ajustado
ajuste2$coefficients ### Estimativas dos parâmetros
##      (Intercept)      perc.alumni I(perc.alumni^2) 
##     45.896244242      1.085947860     -0.007651754
ajuste2$residuals ### Resíduos ordeinários, resultado muito longo, então resumido a seguir
head(ajuste2$residuals) 
## Abilene Christian University           Adelphi University 
##                     2.174234                    -5.312561 
##               Adrian College          Agnes Scott College 
##                   -17.588102                   -16.601064 
##    Alaska Pacific University            Albertson College 
##                   -33.037533                    -1.915809
tail(ajuste2$residuals)
## Worcester Polytechnic Institute         Worcester State College 
##                        8.026956                      -19.599771 
##               Xavier University  Xavier University of Louisiana 
##                       10.792707                      -15.554500 
##                 Yale University    York College of Pennsylvania 
##                       18.264171                       28.696191
ajuste2$fitted.values ### Valores ajustados pelo modelo, resultado muito longo, então resumido a seguir
head(ajuste2$fitted.values) 
## Abilene Christian University           Adelphi University 
##                     57.82577                     61.31256 
##               Adrian College          Agnes Scott College 
##                     71.58810                     75.60106 
##    Alaska Pacific University            Albertson College 
##                     48.03753                     56.91581
tail(ajuste2$fitted.values)
## Worcester Polytechnic Institute         Worcester State College 
##                        73.97304                        59.59977 
##               Xavier University  Xavier University of Louisiana 
##                        72.20729                        64.55450 
##                 Yale University    York College of Pennsylvania 
##                        80.73583                        70.30381
model.matrix(ajuste2) ### Matriz do modelo (matriz X)
##                                               (Intercept) perc.alumni
## Abilene Christian University                            1          12
## Adelphi University                                      1          16
## Adrian College                                          1          30
## Agnes Scott College                                     1          37
## Alaska Pacific University                               1           2
## Albertson College                                       1          11
## Albertus Magnus College                                 1          26
## Albion College                                          1          37
## Albright College                                        1          23
## Alderson-Broaddus College                               1          15
## Alfred University                                       1          31
## Allegheny College                                       1          41
## Allentown Coll. of St. Francis de Sales                 1          21
## Alma College                                            1          32
## Alverno College                                         1          26
## American International College                          1          19
## Amherst College                                         1          63
## Anderson University                                     1          14
## Andrews University                                      1          18
## Angelo State University                                 1           5
## Antioch University                                      1          35
## Appalachian State University                            1          14
## Aquinas College                                         1          25
## Arizona State University Main campus                    1           5
## Arkansas College (Lyon College)                         1          24
## Arkansas Tech University                                1           5
## Assumption College                                      1          30
## Auburn University-Main Campus                           1          18
## Augsburg College                                        1          31
## Augustana College IL                                    1          40
## Augustana College                                       1          30
## Austin College                                          1          33
## Averett College                                         1          11
## Baker University                                        1          21
## Baldwin-Wallace College                                 1          20
## Barat College                                           1          35
## Bard College                                            1          30
## Barnard College                                         1          33
## Barry University                                        1          11
## Baylor University                                       1          38
## Beaver College                                          1          30
## Bellarmine College                                      1          31
## Belmont Abbey College                                   1          10
## Belmont University                                      1          19
## Beloit College                                          1          26
## Bemidji State University                                1          16
## Benedictine College                                     1          18
## Bennington College                                      1          33
## Bentley College                                         1          20
## Berry College                                           1          17
## Bethany College                                         1          29
## Bethel College KS                                       1          32
## Bethel College                                          1          13
## Bethune Cookman College                                 1           9
## Birmingham-Southern College                             1          34
## Blackburn College                                       1          53
## Bloomsburg Univ. of Pennsylvania                        1          19
## Bluefield College                                       1           3
## Bluffton College                                        1          19
## Boston University                                       1          16
## Bowdoin College                                         1          52
## Bowling Green State University                          1          14
## Bradford College                                        1          21
## Bradley University                                      1          21
## Brandeis University                                     1          24
## Brenau University                                       1          12
## Brewton-Parker College                                  1          10
## Briar Cliff College                                     1          26
## Bridgewater College                                     1          24
## Brigham Young University at Provo                       1          40
## Brown University                                        1          39
## Bryn Mawr College                                       1          49
## Bucknell University                                     1          36
## Buena Vista College                                     1          10
## Butler University                                       1          29
## Cabrini College                                         1          36
## Caldwell College                                        1          25
## California Lutheran University                          1          17
## California Polytechnic-San Luis                         1          13
## California State University at Fresno                   1           8
## Calvin College                                          1          41
## Campbell University                                     1          34
## Campbellsville College                                  1          13
## Canisius College                                        1          26
## Capital University                                      1          27
## Capitol College                                         1          24
## Carleton College                                        1          60
## Carnegie Mellon University                              1          31
## Carroll College                                         1          25
## Carson-Newman College                                   1          16
## Carthage College                                        1          22
## Case Western Reserve University                         1          29
## Castleton State College                                 1           8
## Catawba College                                         1          27
## Catholic University of America                          1          18
## Cazenovia College                                       1          20
## Cedar Crest College                                     1          39
## Cedarville College                                      1          34
## Centenary College                                       1          20
## Centenary College of Louisiana                          1          25
## Center for Creative Studies                             1           4
## Central College                                         1          29
## Central Connecticut State University                    1           4
## Central Missouri State University                       1           4
## Central Washington University                           1           0
## Central Wesleyan College                                1          18
## Centre College                                          1          60
## Chapman University                                      1           6
## Chatham College                                         1          37
## Chestnut Hill College                                   1          29
## Christendom College                                     1          17
## Christian Brothers University                           1          24
## Christopher Newport University                          1          16
## Claflin College                                         1          31
## Claremont McKenna College                               1          52
## Clark University                                        1          35
## Clarke College                                          1          27
## Clarkson University                                     1          32
## Clemson University                                      1          17
## Clinch Valley Coll. of  the Univ. of Virginia           1           9
## Coe College                                             1          32
## Coker College                                           1          39
## Colby College                                           1          41
## Colgate University                                      1          45
## College Misericordia                                    1          23
## College of Charleston                                   1          18
## College of Mount St. Joseph                             1          35
## College of Mount St. Vincent                            1          35
## College of Notre Dame                                   1           7
## College of Notre Dame of Maryland                       1          32
## College of Saint Benedict                               1          26
## College of Saint Catherine                              1          32
## College of Saint Elizabeth                              1          23
## College of Saint Rose                                   1          28
## College of Santa Fe                                     1           7
## College of St. Joseph                                   1          19
## College of St. Scholastica                              1          33
## College of the Holy Cross                               1          55
## College of William and Mary                             1          31
## College of Wooster                                      1          43
## Colorado College                                        1          51
## Colorado State University                               1          10
## Columbia College MO                                     1           2
## Columbia College                                        1          34
## Columbia University                                     1          21
## Concordia College at St. Paul                           1          18
## Concordia Lutheran College                              1           9
## Concordia University CA                                 1          13
## Concordia University                                    1          13
## Connecticut College                                     1          40
## Converse College                                        1          31
## Cornell College                                         1          31
## Creighton University                                    1          32
## Culver-Stockton College                                 1          28
## Cumberland College                                      1           4
## D'Youville College                                      1          42
## Dana College                                            1          25
## Daniel Webster College                                  1          10
## Dartmouth College                                       1          49
## Davidson College                                        1          46
## Defiance College                                        1          19
## Delta State University                                  1          16
## Denison University                                      1          45
## DePauw University                                       1          31
## Dickinson College                                       1          39
## Dickinson State University                              1          28
## Dillard University                                      1          12
## Doane College                                           1          42
## Dominican College of Blauvelt                           1           5
## Dordt College                                           1          17
## Dowling College                                         1           7
## Drake University                                        1          24
## Drew University                                         1          28
## Drury College                                           1          35
## Duke University                                         1          44
## Earlham College                                         1          46
## East Carolina University                                1          18
## East Tennessee State University                         1           9
## East Texas Baptist University                           1           7
## Eastern College                                         1          22
## Eastern Connecticut State University                    1          14
## Eastern Illinois University                             1           5
## Eastern Mennonite College                               1          29
## Eastern Nazarene College                                1          17
## Eckerd College                                          1          26
## Elizabethtown College                                   1          25
## Elmira College                                          1          21
## Elms College                                            1          21
## Elon College                                            1          34
## Embry Riddle Aeronautical University                    1           4
## Emory & Henry College                                   1          51
## Emory University                                        1          28
## Emporia State University                                1           4
## Erskine College                                         1          47
## Eureka College                                          1          31
## Evergreen State College                                 1          14
## Fairfield University                                    1          30
## Fayetteville State University                           1          10
## Ferrum College                                          1           9
## Flagler College                                         1           9
## Florida Institute of Technology                         1           7
## Florida International University                        1          20
## Florida Southern College                                1          10
## Florida State University                                1          15
## Fontbonne College                                       1          24
## Fordham University                                      1          14
## Fort Lewis College                                      1           6
## Francis Marion University                               1           8
## Franciscan University of Steubenville                   1           8
## Franklin College                                        1          37
## Franklin Pierce College                                 1          16
## Freed-Hardeman University                               1          13
## Fresno Pacific College                                  1          14
## Furman University                                       1          28
## Gannon University                                       1          18
## Gardner Webb University                                 1          12
## Geneva College                                          1          26
## George Fox College                                      1          22
## George Mason University                                 1           7
## George Washington University                            1          15
## Georgetown College                                      1          28
## Georgetown University                                   1          27
## Georgia Institute of Technology                         1          33
## Georgia State University                                1          10
## Georgian Court College                                  1          27
## Gettysburg College                                      1          32
## Goldey Beacom College                                   1           4
## Gonzaga University                                      1          32
## Gordon College                                          1          32
## Goshen College                                          1          46
## Goucher College                                         1          34
## Grace College and Seminary                              1          26
## Graceland College                                       1          24
## Grand Valley State University                           1           9
## Green Mountain College                                  1          24
## Greensboro College                                      1          31
## Greenville College                                      1          16
## Grinnell College                                        1          54
## Grove City College                                      1          18
## Guilford College                                        1          30
## Gustavus Adolphus College                               1          58
## Gwynedd Mercy College                                   1          22
## Hamilton College                                        1          60
## Hamline University                                      1          33
## Hampden - Sydney College                                1          53
## Hampton University                                      1           9
## Hanover College                                         1          26
## Hardin-Simmons University                               1          10
## Harding University                                      1          37
## Hartwick College                                        1          32
## Harvard University                                      1          52
## Harvey Mudd College                                     1          46
## Hastings College                                        1          17
## Hendrix College                                         1          26
## Hillsdale College                                       1          31
## Hiram College                                           1          34
## Hobart and William Smith Colleges                       1          37
## Hofstra University                                      1          10
## Hollins College                                         1          48
## Hood College                                            1          34
## Hope College                                            1          40
## Houghton College                                        1          24
## Huntingdon College                                      1           9
## Huntington College                                      1          25
## Huron University                                        1           4
## Husson College                                          1           4
## Illinois Benedictine College                            1          29
## Illinois College                                        1          30
## Illinois Institute of Technology                        1          26
## Illinois State University                               1          16
## Illinois Wesleyan University                            1          34
## Immaculata College                                      1          33
## Incarnate Word College                                  1          21
## Indiana State University                                1           8
## Indiana University at Bloomington                       1          24
## Indiana Wesleyan University                             1          15
## Iona College                                            1          14
## Iowa State University                                   1          22
## Ithaca College                                          1          25
## James Madison University                                1          29
## Jamestown College                                       1          21
## Jersey City State College                               1          10
## John Brown University                                   1          19
## John Carroll University                                 1          28
## Johns Hopkins University                                1          38
## Johnson State College                                   1          15
## Judson College                                          1          30
## Juniata College                                         1          37
## Kansas State University                                 1          22
## Kansas Wesleyan University                              1          14
## Keene State College                                     1          13
## Kentucky Wesleyan College                               1          32
## Kenyon College                                          1          46
## Keuka College                                           1          43
## King's College                                          1          37
## King College                                            1          25
## Knox College                                            1          33
## La Roche College                                        1          14
## La Salle University                                     1           9
## Lafayette College                                       1          38
## LaGrange College                                        1          12
## Lake Forest College                                     1          19
## Lakeland College                                        1          25
## Lamar University                                        1          12
## Lambuth University                                      1          10
## Lander University                                       1          11
## Lawrence University                                     1          57
## Le Moyne College                                        1          28
## Lebanon Valley College                                  1          30
## Lehigh University                                       1          43
## Lenoir-Rhyne College                                    1          20
## Lesley College                                          1          18
## LeTourneau University                                   1          23
## Lewis and Clark College                                 1          21
## Lewis University                                        1          10
## Lincoln Memorial University                             1          35
## Lincoln University                                      1           8
## Lindenwood College                                      1           9
## Linfield College                                        1          34
## Livingstone College                                     1          16
## Lock Haven University of Pennsylvania                   1          14
## Longwood College                                        1          23
## Loras College                                           1          24
## Louisiana College                                       1          11
## Louisiana State University at Baton Rouge               1          11
## Louisiana Tech University                               1          13
## Loyola College                                          1          27
## Loyola Marymount University                             1          10
## Loyola University                                       1          14
## Loyola University Chicago                               1          15
## Luther College                                          1          38
## Lycoming College                                        1          32
## Lynchburg College                                       1          24
## Lyndon State College                                    1          15
## Macalester College                                      1          37
## MacMurray College                                       1          33
## Malone College                                          1          16
## Manchester College                                      1          20
## Manhattan College                                       1          25
## Manhattanville College                                  1          24
## Mankato State University                                1          11
## Marian College of Fond du Lac                           1          21
## Marietta College                                        1          30
## Marist College                                          1          34
## Marquette University                                    1          25
## Marshall University                                     1          10
## Mary Baldwin College                                    1          50
## Mary Washington College                                 1          30
## Marymount College Tarrytown                             1          30
## Marymount Manhattan College                             1          20
## Marymount University                                    1          17
## Maryville College                                       1          43
## Maryville University                                    1          13
## Marywood College                                        1          30
## Massachusetts Institute of Technology                   1          35
## Mayville State University                               1          11
## McKendree College                                       1          21
## McMurry University                                      1          11
## McPherson College                                       1          45
## Mercer University                                       1          15
## Mercyhurst College                                      1          29
## Meredith College                                        1          33
## Merrimack College                                       1          22
## Mesa State College                                      1          12
## Messiah College                                         1          30
## Miami University at Oxford                              1          20
## Michigan State University                               1           9
## Michigan Technological University                       1          25
## MidAmerica Nazarene College                             1          20
## Millersville University of Penn.                        1          20
## Milligan College                                        1          16
## Millikin University                                     1          25
## Millsaps College                                        1          38
## Milwaukee School of Engineering                         1          23
## Mississippi College                                     1          18
## Mississippi State University                            1          20
## Mississippi University for Women                        1           8
## Missouri Southern State College                         1           9
## Missouri Valley College                                 1          16
## Monmouth College IL                                     1          43
## Monmouth College                                        1          15
## Montana College of Mineral Sci. & Tech.                 1          31
## Montana State University                                1           8
## Montclair State University                              1           9
## Montreat-Anderson College                               1           4
## Moorhead State University                               1          19
## Moravian College                                        1          28
## Morehouse College                                       1          10
## Morningside College                                     1          32
## Morris College                                          1          34
## Mount Holyoke College                                   1          51
## Mount Marty College                                     1          38
## Mount Mary College                                      1          26
## Mount Mercy College                                     1          30
## Mount Saint Clare College                               1          43
## Mount Saint Mary's College                              1          36
## Mount Saint Mary College                                1          23
## Mount St. Mary's College                                1          19
## Mount Union College                                     1          35
## Mount Vernon Nazarene College                           1           7
## Muhlenberg College                                      1          39
## Murray State University                                 1          27
## Muskingum College                                       1          27
## National-Louis University                               1           2
## Nazareth College of Rochester                           1          24
## New Jersey Institute of Technology                      1          19
## New Mexico Institute of Mining and Tech.                1          11
## New York University                                     1          16
## Newberry College                                        1          32
## Niagara University                                      1          20
## North Adams State College                               1          17
## North Carolina A. & T. State University                 1           9
## North Carolina State University at Raleigh              1          21
## North Carolina Wesleyan College                         1          11
## North Central College                                   1          33
## North Dakota State University                           1          24
## North Park College                                      1          24
## Northeast Missouri State University                     1          13
## Northeastern University                                 1          17
## Northern Arizona University                             1           7
## Northern Illinois University                            1          11
## Northwest Missouri State University                     1          23
## Northwest Nazarene College                              1          20
## Northwestern College                                    1          34
## Northwestern University                                 1          25
## Norwich University                                      1          22
## Notre Dame College                                      1          26
## Oakland University                                      1          13
## Oberlin College                                         1          47
## Occidental College                                      1          30
## Oglethorpe University                                   1          27
## Ohio Northern University                                1          31
## Ohio University                                         1          13
## Ohio Wesleyan University                                1          32
## Oklahoma Baptist University                             1          18
## Oklahoma Christian University                           1           8
## Oklahoma State University                               1          14
## Otterbein College                                       1          30
## Ouachita Baptist University                             1          10
## Our Lady of the Lake University                         1          25
## Pace University                                         1          10
## Pacific Lutheran University                             1          23
## Pacific Union College                                   1          12
## Pacific University                                      1          22
## Pembroke State University                               1           5
## Pennsylvania State Univ. Main Campus                    1          19
## Pepperdine University                                   1          13
## Peru State College                                      1          24
## Pfeiffer College                                        1          13
## Philadelphia Coll. of Textiles and Sci.                 1          15
## Phillips University                                     1          19
## Piedmont College                                        1          17
## Pikeville College                                       1          14
## Pitzer College                                          1          11
## Point Loma Nazarene College                             1          19
## Point Park College                                      1          10
## Polytechnic University                                  1          14
## Prairie View A. and M. University                       1           1
## Presbyterian College                                    1          42
## Princeton University                                    1          54
## Providence College                                      1          35
## Purdue University at West Lafayette                     1          15
## Queens College                                          1          36
## Quincy University                                       1          32
## Quinnipiac College                                      1          33
## Radford University                                      1           9
## Ramapo College of New Jersey                            1           8
## Randolph-Macon College                                  1          38
## Randolph-Macon Woman's College                          1          24
## Reed College                                            1          37
## Regis College                                           1          37
## Rensselaer Polytechnic Institute                        1          21
## Rhodes College                                          1          47
## Rider University                                        1          23
## Ripon College                                           1          49
## Rivier College                                          1          19
## Roanoke College                                         1          26
## Rockhurst College                                       1          21
## Rocky Mountain College                                  1          27
## Roger Williams University                               1           8
## Rollins College                                         1          23
## Rosary College                                          1          30
## Rowan College of New Jersey                             1           6
## Rutgers at New Brunswick                                1          19
## Rutgers State University at Camden                      1          12
## Rutgers State University at Newark                      1          16
## Sacred Heart University                                 1          16
## Saint Ambrose University                                1          16
## Saint Anselm College                                    1          29
## Saint Cloud State University                            1          10
## Saint Francis College IN                                1           4
## Saint Francis College                                   1          24
## Saint John's University                                 1          38
## Saint Joseph's College IN                               1          19
## Saint Joseph's College                                  1           8
## Saint Joseph's University                               1          13
## Saint Joseph College                                    1          32
## Saint Louis University                                  1          19
## Saint Mary's College                                    1          31
## Saint Mary's College of Minnesota                       1          19
## Saint Mary-of-the-Woods College                         1          37
## Saint Michael's College                                 1          34
## Saint Olaf College                                      1          31
## Saint Peter's College                                   1          22
## Saint Vincent College                                   1          31
## Saint Xavier University                                 1          15
## Salem-Teikyo University                                 1           9
## Salem College                                           1          46
## Salisbury State University                              1          18
## Samford University                                      1          17
## San Diego State University                              1           7
## Santa Clara University                                  1          19
## Sarah Lawrence College                                  1          18
## Savannah Coll. of Art and Design                        1          26
## Schreiner College                                       1          23
## Scripps College                                         1          41
## Seattle Pacific University                              1          20
## Seattle University                                      1          16
## Seton Hall University                                   1          15
## Seton Hill College                                      1          37
## Shippensburg University of Penn.                        1          13
## Shorter College                                         1          18
## Siena College                                           1          42
## Siena Heights College                                   1          17
## Simmons College                                         1          33
## Simpson College                                         1          36
## Sioux Falls College                                     1           7
## Skidmore College                                        1          29
## Smith College                                           1          44
## South Dakota State University                           1          29
## Southeast Missouri State University                     1           8
## Southeastern Oklahoma State Univ.                       1           9
## Southern California College                             1          11
## Southern Illinois University at Edwardsville            1           8
## Southern Methodist University                           1          17
## Southwest Baptist University                            1          13
## Southwest Missouri State University                     1          11
## Southwest State University                              1          31
## Southwestern Adventist College                          1          12
## Southwestern College                                    1          12
## Southwestern University                                 1          35
## Spalding University                                     1          40
## Spelman College                                         1          18
## Spring Arbor College                                    1           9
## St. Bonaventure University                              1          32
## St. John's College                                      1          26
## St. John Fisher College                                 1          29
## St. Lawrence University                                 1          38
## St. Martin's College                                    1           8
## St. Mary's College of California                        1          17
## St. Mary's College of Maryland                          1          23
## St. Mary's University of San Antonio                    1           7
## St. Norbert College                                     1          36
## St. Paul's College                                      1           8
## St. Thomas Aquinas College                              1          13
## Stephens College                                        1          17
## Stetson University                                      1          24
## Stevens Institute of Technology                         1          33
## Stockton College of New Jersey                          1           7
## Stonehill College                                       1          30
## SUNY at Albany                                          1          16
## SUNY at Binghamton                                      1          15
## SUNY at Buffalo                                         1          15
## SUNY at Stony Brook                                     1           7
## SUNY College  at Brockport                              1          14
## SUNY College  at Oswego                                 1          21
## SUNY College at Buffalo                                 1          12
## SUNY College at Cortland                                1          17
## SUNY College at Fredonia                                1          10
## SUNY College at Geneseo                                 1          25
## SUNY College at New Paltz                               1           8
## SUNY College at Plattsburgh                             1          16
## SUNY College at Potsdam                                 1          17
## SUNY College at Purchase                                1           8
## Susquehanna University                                  1          37
## Sweet Briar College                                     1          48
## Syracuse University                                     1          13
## Tabor College                                           1          15
## Talladega College                                       1           7
## Taylor University                                       1          32
## Tennessee Wesleyan College                              1          16
## Texas A&M Univ. at College Station                      1          29
## Texas A&M University at Galveston                       1          16
## Texas Christian University                              1          23
## Texas Lutheran College                                  1          24
## Texas Southern University                               1          21
## Texas Wesleyan University                               1          10
## The Citadel                                             1          17
## Thiel College                                           1          16
## Tiffin University                                       1          40
## Transylvania University                                 1          41
## Trenton State College                                   1           6
## Tri-State University                                    1          24
## Trinity College CT                                      1          48
## Trinity College DC                                      1          37
## Trinity College VT                                      1          26
## Trinity University                                      1          20
## Tulane University                                       1          21
## Tusculum College                                        1          28
## Tuskegee University                                     1           7
## Union College KY                                        1           9
## Union College NY                                        1          49
## Univ. of Wisconsin at OshKosh                           1          14
## University of Alabama at Birmingham                     1          16
## University of Arkansas at Fayetteville                  1          10
## University of California at Berkeley                    1          10
## University of California at Irvine                      1          11
## University of Central Florida                           1           9
## University of Charleston                                1          10
## University of Chicago                                   1          36
## University of Cincinnati                                1           6
## University of Connecticut at Storrs                     1          16
## University of Dallas                                    1          26
## University of Dayton                                    1          25
## University of Delaware                                  1          15
## University of Denver                                    1          21
## University of Detroit Mercy                             1          14
## University of Dubuque                                   1          18
## University of Evansville                                1          26
## University of Florida                                   1          20
## University of Georgia                                   1          22
## University of Hartford                                  1           9
## University of Hawaii at Manoa                           1           6
## University of Illinois - Urbana                         1          13
## University of Illinois at Chicago                       1           6
## University of Indianapolis                              1          23
## University of Kansas                                    1          17
## University of La Verne                                  1          23
## University of Louisville                                1          24
## University of Maine at Farmington                       1          26
## University of Maine at Machias                          1           4
## University of Maine at Presque Isle                     1          11
## University of Maryland at Baltimore County              1           6
## University of Maryland at College Park                  1          12
## University of Massachusetts at Amherst                  1          15
## University of Massachusetts at Dartmouth                1          20
## University of Miami                                     1          17
## University of Michigan at Ann Arbor                     1          26
## University of Minnesota at Duluth                       1          11
## University of Minnesota at Morris                       1          16
## University of Minnesota Twin Cities                     1          37
## University of Mississippi                               1          14
## University of Missouri at Columbia                      1          15
## University of Missouri at Rolla                         1          23
## University of Missouri at Saint Louis                   1          15
## University of Mobile                                    1           4
## University of Montevallo                                1           8
## University of Nebraska at Lincoln                       1          48
## University of New England                               1          13
## University of New Hampshire                             1          16
## University of North Carolina at Asheville               1          11
## University of North Carolina at Chapel Hill             1          23
## University of North Carolina at Charlotte               1           7
## University of North Carolina at Greensboro              1          17
## University of North Carolina at Wilmington              1          15
## University of North Dakota                              1          16
## University of North Florida                             1          14
## University of North Texas                               1           6
## University of Northern Colorado                         1           8
## University of Northern Iowa                             1          26
## University of Notre Dame                                1          46
## University of Oklahoma                                  1          11
## University of Oregon                                    1          13
## University of Pennsylvania                              1          38
## University of Pittsburgh-Main Campus                    1          10
## University of Portland                                  1          17
## University of Puget Sound                               1          17
## University of Rhode Island                              1           7
## University of Richmond                                  1          32
## University of Rochester                                 1          23
## University of San Diego                                 1          13
## University of San Francisco                             1           8
## University of Sci. and Arts of Oklahoma                 1           3
## University of Scranton                                  1          41
## University of South Carolina at Aiken                   1           3
## University of South Carolina at Columbia                1          18
## University of South Florida                             1           7
## University of Southern California                       1          10
## University of Southern Colorado                         1           0
## University of Southern Indiana                          1          21
## University of Southern Mississippi                      1          23
## University of St. Thomas MN                             1          13
## University of St. Thomas TX                             1          17
## University of Tennessee at Knoxville                    1          22
## University of Texas at Arlington                        1           4
## University of Texas at Austin                           1          11
## University of Texas at San Antonio                      1           3
## University of the Arts                                  1           9
## University of the Pacific                               1          14
## University of the South                                 1          52
## University of Tulsa                                     1          10
## University of Utah                                      1           9
## University of Vermont                                   1          10
## University of Virginia                                  1          22
## University of Washington                                1          10
## University of West Florida                              1          12
## University of Wisconsin-Stout                           1          17
## University of Wisconsin-Superior                        1          15
## University of Wisconsin-Whitewater                      1          16
## University of Wisconsin at Green Bay                    1           1
## University of Wisconsin at Madison                      1          20
## University of Wisconsin at Milwaukee                    1           8
## University of Wyoming                                   1          13
## Upper Iowa University                                   1          19
## Ursinus College                                         1          40
## Ursuline College                                        1          15
## Valley City State University                            1          25
## Valparaiso University                                   1          23
## Vanderbilt University                                   1          26
## Vassar College                                          1          39
## Villanova University                                    1          24
## Virginia Commonwealth University                        1          11
## Virginia State University                               1          11
## Virginia Tech                                           1          20
## Virginia Union University                               1           8
## Virginia Wesleyan College                               1          14
## Viterbo College                                         1          31
## Voorhees College                                        1           3
## Wabash College                                          1          55
## Wagner College                                          1          23
## Wake Forest University                                  1          37
## Walsh University                                        1          33
## Warren Wilson College                                   1          20
## Wartburg College                                        1          37
## Washington and Jefferson College                        1          40
## Washington and Lee University                           1          45
## Washington College                                      1          37
## Washington State University                             1          30
## Washington University                                   1          31
## Wayne State College                                     1          29
## Waynesburg College                                      1          26
## Webber College                                          1           4
## Webster University                                      1          14
## Wellesley College                                       1          51
## Wells College                                           1          42
## Wentworth Institute of Technology                       1           8
## Wesley College                                          1          15
## Wesleyan University                                     1          39
## West Chester University of Penn.                        1          16
## West Liberty State College                              1          10
## West Virginia Wesleyan College                          1          42
## Western Carolina University                             1           9
## Western Maryland College                                1          39
## Western Michigan University                             1          11
## Western New England College                             1          15
## Western State College of Colorado                       1           4
## Western Washington University                           1          10
## Westfield State College                                 1          20
## Westminster College MO                                  1          20
## Westminster College                                     1          41
## Westminster College of Salt Lake City                   1          34
## Westmont College                                        1          17
## Wheaton College IL                                      1          40
## Westminster College PA                                  1          41
## Wheeling Jesuit College                                 1          27
## Whitman College                                         1          51
## Whittier College                                        1          29
## Whitworth College                                       1          20
## Widener University                                      1          19
## Wilkes University                                       1          24
## Willamette University                                   1          37
## William Jewell College                                  1          19
## William Woods University                                1          16
## Williams College                                        1          64
## Wilson College                                          1          43
## Wingate College                                         1           8
## Winona State University                                 1          18
## Winthrop University                                     1          26
## Wisconsin Lutheran College                              1          26
## Wittenberg University                                   1          29
## Wofford College                                         1          42
## Worcester Polytechnic Institute                         1          34
## Worcester State College                                 1          14
## Xavier University                                       1          31
## Xavier University of Louisiana                          1          20
## Yale University                                         1          49
## York College of Pennsylvania                            1          28
##                                               I(perc.alumni^2)
## Abilene Christian University                               144
## Adelphi University                                         256
## Adrian College                                             900
## Agnes Scott College                                       1369
## Alaska Pacific University                                    4
## Albertson College                                          121
## Albertus Magnus College                                    676
## Albion College                                            1369
## Albright College                                           529
## Alderson-Broaddus College                                  225
## Alfred University                                          961
## Allegheny College                                         1681
## Allentown Coll. of St. Francis de Sales                    441
## Alma College                                              1024
## Alverno College                                            676
## American International College                             361
## Amherst College                                           3969
## Anderson University                                        196
## Andrews University                                         324
## Angelo State University                                     25
## Antioch University                                        1225
## Appalachian State University                               196
## Aquinas College                                            625
## Arizona State University Main campus                        25
## Arkansas College (Lyon College)                            576
## Arkansas Tech University                                    25
## Assumption College                                         900
## Auburn University-Main Campus                              324
## Augsburg College                                           961
## Augustana College IL                                      1600
## Augustana College                                          900
## Austin College                                            1089
## Averett College                                            121
## Baker University                                           441
## Baldwin-Wallace College                                    400
## Barat College                                             1225
## Bard College                                               900
## Barnard College                                           1089
## Barry University                                           121
## Baylor University                                         1444
## Beaver College                                             900
## Bellarmine College                                         961
## Belmont Abbey College                                      100
## Belmont University                                         361
## Beloit College                                             676
## Bemidji State University                                   256
## Benedictine College                                        324
## Bennington College                                        1089
## Bentley College                                            400
## Berry College                                              289
## Bethany College                                            841
## Bethel College KS                                         1024
## Bethel College                                             169
## Bethune Cookman College                                     81
## Birmingham-Southern College                               1156
## Blackburn College                                         2809
## Bloomsburg Univ. of Pennsylvania                           361
## Bluefield College                                            9
## Bluffton College                                           361
## Boston University                                          256
## Bowdoin College                                           2704
## Bowling Green State University                             196
## Bradford College                                           441
## Bradley University                                         441
## Brandeis University                                        576
## Brenau University                                          144
## Brewton-Parker College                                     100
## Briar Cliff College                                        676
## Bridgewater College                                        576
## Brigham Young University at Provo                         1600
## Brown University                                          1521
## Bryn Mawr College                                         2401
## Bucknell University                                       1296
## Buena Vista College                                        100
## Butler University                                          841
## Cabrini College                                           1296
## Caldwell College                                           625
## California Lutheran University                             289
## California Polytechnic-San Luis                            169
## California State University at Fresno                       64
## Calvin College                                            1681
## Campbell University                                       1156
## Campbellsville College                                     169
## Canisius College                                           676
## Capital University                                         729
## Capitol College                                            576
## Carleton College                                          3600
## Carnegie Mellon University                                 961
## Carroll College                                            625
## Carson-Newman College                                      256
## Carthage College                                           484
## Case Western Reserve University                            841
## Castleton State College                                     64
## Catawba College                                            729
## Catholic University of America                             324
## Cazenovia College                                          400
## Cedar Crest College                                       1521
## Cedarville College                                        1156
## Centenary College                                          400
## Centenary College of Louisiana                             625
## Center for Creative Studies                                 16
## Central College                                            841
## Central Connecticut State University                        16
## Central Missouri State University                           16
## Central Washington University                                0
## Central Wesleyan College                                   324
## Centre College                                            3600
## Chapman University                                          36
## Chatham College                                           1369
## Chestnut Hill College                                      841
## Christendom College                                        289
## Christian Brothers University                              576
## Christopher Newport University                             256
## Claflin College                                            961
## Claremont McKenna College                                 2704
## Clark University                                          1225
## Clarke College                                             729
## Clarkson University                                       1024
## Clemson University                                         289
## Clinch Valley Coll. of  the Univ. of Virginia               81
## Coe College                                               1024
## Coker College                                             1521
## Colby College                                             1681
## Colgate University                                        2025
## College Misericordia                                       529
## College of Charleston                                      324
## College of Mount St. Joseph                               1225
## College of Mount St. Vincent                              1225
## College of Notre Dame                                       49
## College of Notre Dame of Maryland                         1024
## College of Saint Benedict                                  676
## College of Saint Catherine                                1024
## College of Saint Elizabeth                                 529
## College of Saint Rose                                      784
## College of Santa Fe                                         49
## College of St. Joseph                                      361
## College of St. Scholastica                                1089
## College of the Holy Cross                                 3025
## College of William and Mary                                961
## College of Wooster                                        1849
## Colorado College                                          2601
## Colorado State University                                  100
## Columbia College MO                                          4
## Columbia College                                          1156
## Columbia University                                        441
## Concordia College at St. Paul                              324
## Concordia Lutheran College                                  81
## Concordia University CA                                    169
## Concordia University                                       169
## Connecticut College                                       1600
## Converse College                                           961
## Cornell College                                            961
## Creighton University                                      1024
## Culver-Stockton College                                    784
## Cumberland College                                          16
## D'Youville College                                        1764
## Dana College                                               625
## Daniel Webster College                                     100
## Dartmouth College                                         2401
## Davidson College                                          2116
## Defiance College                                           361
## Delta State University                                     256
## Denison University                                        2025
## DePauw University                                          961
## Dickinson College                                         1521
## Dickinson State University                                 784
## Dillard University                                         144
## Doane College                                             1764
## Dominican College of Blauvelt                               25
## Dordt College                                              289
## Dowling College                                             49
## Drake University                                           576
## Drew University                                            784
## Drury College                                             1225
## Duke University                                           1936
## Earlham College                                           2116
## East Carolina University                                   324
## East Tennessee State University                             81
## East Texas Baptist University                               49
## Eastern College                                            484
## Eastern Connecticut State University                       196
## Eastern Illinois University                                 25
## Eastern Mennonite College                                  841
## Eastern Nazarene College                                   289
## Eckerd College                                             676
## Elizabethtown College                                      625
## Elmira College                                             441
## Elms College                                               441
## Elon College                                              1156
## Embry Riddle Aeronautical University                        16
## Emory & Henry College                                     2601
## Emory University                                           784
## Emporia State University                                    16
## Erskine College                                           2209
## Eureka College                                             961
## Evergreen State College                                    196
## Fairfield University                                       900
## Fayetteville State University                              100
## Ferrum College                                              81
## Flagler College                                             81
## Florida Institute of Technology                             49
## Florida International University                           400
## Florida Southern College                                   100
## Florida State University                                   225
## Fontbonne College                                          576
## Fordham University                                         196
## Fort Lewis College                                          36
## Francis Marion University                                   64
## Franciscan University of Steubenville                       64
## Franklin College                                          1369
## Franklin Pierce College                                    256
## Freed-Hardeman University                                  169
## Fresno Pacific College                                     196
## Furman University                                          784
## Gannon University                                          324
## Gardner Webb University                                    144
## Geneva College                                             676
## George Fox College                                         484
## George Mason University                                     49
## George Washington University                               225
## Georgetown College                                         784
## Georgetown University                                      729
## Georgia Institute of Technology                           1089
## Georgia State University                                   100
## Georgian Court College                                     729
## Gettysburg College                                        1024
## Goldey Beacom College                                       16
## Gonzaga University                                        1024
## Gordon College                                            1024
## Goshen College                                            2116
## Goucher College                                           1156
## Grace College and Seminary                                 676
## Graceland College                                          576
## Grand Valley State University                               81
## Green Mountain College                                     576
## Greensboro College                                         961
## Greenville College                                         256
## Grinnell College                                          2916
## Grove City College                                         324
## Guilford College                                           900
## Gustavus Adolphus College                                 3364
## Gwynedd Mercy College                                      484
## Hamilton College                                          3600
## Hamline University                                        1089
## Hampden - Sydney College                                  2809
## Hampton University                                          81
## Hanover College                                            676
## Hardin-Simmons University                                  100
## Harding University                                        1369
## Hartwick College                                          1024
## Harvard University                                        2704
## Harvey Mudd College                                       2116
## Hastings College                                           289
## Hendrix College                                            676
## Hillsdale College                                          961
## Hiram College                                             1156
## Hobart and William Smith Colleges                         1369
## Hofstra University                                         100
## Hollins College                                           2304
## Hood College                                              1156
## Hope College                                              1600
## Houghton College                                           576
## Huntingdon College                                          81
## Huntington College                                         625
## Huron University                                            16
## Husson College                                              16
## Illinois Benedictine College                               841
## Illinois College                                           900
## Illinois Institute of Technology                           676
## Illinois State University                                  256
## Illinois Wesleyan University                              1156
## Immaculata College                                        1089
## Incarnate Word College                                     441
## Indiana State University                                    64
## Indiana University at Bloomington                          576
## Indiana Wesleyan University                                225
## Iona College                                               196
## Iowa State University                                      484
## Ithaca College                                             625
## James Madison University                                   841
## Jamestown College                                          441
## Jersey City State College                                  100
## John Brown University                                      361
## John Carroll University                                    784
## Johns Hopkins University                                  1444
## Johnson State College                                      225
## Judson College                                             900
## Juniata College                                           1369
## Kansas State University                                    484
## Kansas Wesleyan University                                 196
## Keene State College                                        169
## Kentucky Wesleyan College                                 1024
## Kenyon College                                            2116
## Keuka College                                             1849
## King's College                                            1369
## King College                                               625
## Knox College                                              1089
## La Roche College                                           196
## La Salle University                                         81
## Lafayette College                                         1444
## LaGrange College                                           144
## Lake Forest College                                        361
## Lakeland College                                           625
## Lamar University                                           144
## Lambuth University                                         100
## Lander University                                          121
## Lawrence University                                       3249
## Le Moyne College                                           784
## Lebanon Valley College                                     900
## Lehigh University                                         1849
## Lenoir-Rhyne College                                       400
## Lesley College                                             324
## LeTourneau University                                      529
## Lewis and Clark College                                    441
## Lewis University                                           100
## Lincoln Memorial University                               1225
## Lincoln University                                          64
## Lindenwood College                                          81
## Linfield College                                          1156
## Livingstone College                                        256
## Lock Haven University of Pennsylvania                      196
## Longwood College                                           529
## Loras College                                              576
## Louisiana College                                          121
## Louisiana State University at Baton Rouge                  121
## Louisiana Tech University                                  169
## Loyola College                                             729
## Loyola Marymount University                                100
## Loyola University                                          196
## Loyola University Chicago                                  225
## Luther College                                            1444
## Lycoming College                                          1024
## Lynchburg College                                          576
## Lyndon State College                                       225
## Macalester College                                        1369
## MacMurray College                                         1089
## Malone College                                             256
## Manchester College                                         400
## Manhattan College                                          625
## Manhattanville College                                     576
## Mankato State University                                   121
## Marian College of Fond du Lac                              441
## Marietta College                                           900
## Marist College                                            1156
## Marquette University                                       625
## Marshall University                                        100
## Mary Baldwin College                                      2500
## Mary Washington College                                    900
## Marymount College Tarrytown                                900
## Marymount Manhattan College                                400
## Marymount University                                       289
## Maryville College                                         1849
## Maryville University                                       169
## Marywood College                                           900
## Massachusetts Institute of Technology                     1225
## Mayville State University                                  121
## McKendree College                                          441
## McMurry University                                         121
## McPherson College                                         2025
## Mercer University                                          225
## Mercyhurst College                                         841
## Meredith College                                          1089
## Merrimack College                                          484
## Mesa State College                                         144
## Messiah College                                            900
## Miami University at Oxford                                 400
## Michigan State University                                   81
## Michigan Technological University                          625
## MidAmerica Nazarene College                                400
## Millersville University of Penn.                           400
## Milligan College                                           256
## Millikin University                                        625
## Millsaps College                                          1444
## Milwaukee School of Engineering                            529
## Mississippi College                                        324
## Mississippi State University                               400
## Mississippi University for Women                            64
## Missouri Southern State College                             81
## Missouri Valley College                                    256
## Monmouth College IL                                       1849
## Monmouth College                                           225
## Montana College of Mineral Sci. & Tech.                    961
## Montana State University                                    64
## Montclair State University                                  81
## Montreat-Anderson College                                   16
## Moorhead State University                                  361
## Moravian College                                           784
## Morehouse College                                          100
## Morningside College                                       1024
## Morris College                                            1156
## Mount Holyoke College                                     2601
## Mount Marty College                                       1444
## Mount Mary College                                         676
## Mount Mercy College                                        900
## Mount Saint Clare College                                 1849
## Mount Saint Mary's College                                1296
## Mount Saint Mary College                                   529
## Mount St. Mary's College                                   361
## Mount Union College                                       1225
## Mount Vernon Nazarene College                               49
## Muhlenberg College                                        1521
## Murray State University                                    729
## Muskingum College                                          729
## National-Louis University                                    4
## Nazareth College of Rochester                              576
## New Jersey Institute of Technology                         361
## New Mexico Institute of Mining and Tech.                   121
## New York University                                        256
## Newberry College                                          1024
## Niagara University                                         400
## North Adams State College                                  289
## North Carolina A. & T. State University                     81
## North Carolina State University at Raleigh                 441
## North Carolina Wesleyan College                            121
## North Central College                                     1089
## North Dakota State University                              576
## North Park College                                         576
## Northeast Missouri State University                        169
## Northeastern University                                    289
## Northern Arizona University                                 49
## Northern Illinois University                               121
## Northwest Missouri State University                        529
## Northwest Nazarene College                                 400
## Northwestern College                                      1156
## Northwestern University                                    625
## Norwich University                                         484
## Notre Dame College                                         676
## Oakland University                                         169
## Oberlin College                                           2209
## Occidental College                                         900
## Oglethorpe University                                      729
## Ohio Northern University                                   961
## Ohio University                                            169
## Ohio Wesleyan University                                  1024
## Oklahoma Baptist University                                324
## Oklahoma Christian University                               64
## Oklahoma State University                                  196
## Otterbein College                                          900
## Ouachita Baptist University                                100
## Our Lady of the Lake University                            625
## Pace University                                            100
## Pacific Lutheran University                                529
## Pacific Union College                                      144
## Pacific University                                         484
## Pembroke State University                                   25
## Pennsylvania State Univ. Main Campus                       361
## Pepperdine University                                      169
## Peru State College                                         576
## Pfeiffer College                                           169
## Philadelphia Coll. of Textiles and Sci.                    225
## Phillips University                                        361
## Piedmont College                                           289
## Pikeville College                                          196
## Pitzer College                                             121
## Point Loma Nazarene College                                361
## Point Park College                                         100
## Polytechnic University                                     196
## Prairie View A. and M. University                            1
## Presbyterian College                                      1764
## Princeton University                                      2916
## Providence College                                        1225
## Purdue University at West Lafayette                        225
## Queens College                                            1296
## Quincy University                                         1024
## Quinnipiac College                                        1089
## Radford University                                          81
## Ramapo College of New Jersey                                64
## Randolph-Macon College                                    1444
## Randolph-Macon Woman's College                             576
## Reed College                                              1369
## Regis College                                             1369
## Rensselaer Polytechnic Institute                           441
## Rhodes College                                            2209
## Rider University                                           529
## Ripon College                                             2401
## Rivier College                                             361
## Roanoke College                                            676
## Rockhurst College                                          441
## Rocky Mountain College                                     729
## Roger Williams University                                   64
## Rollins College                                            529
## Rosary College                                             900
## Rowan College of New Jersey                                 36
## Rutgers at New Brunswick                                   361
## Rutgers State University at Camden                         144
## Rutgers State University at Newark                         256
## Sacred Heart University                                    256
## Saint Ambrose University                                   256
## Saint Anselm College                                       841
## Saint Cloud State University                               100
## Saint Francis College IN                                    16
## Saint Francis College                                      576
## Saint John's University                                   1444
## Saint Joseph's College IN                                  361
## Saint Joseph's College                                      64
## Saint Joseph's University                                  169
## Saint Joseph College                                      1024
## Saint Louis University                                     361
## Saint Mary's College                                       961
## Saint Mary's College of Minnesota                          361
## Saint Mary-of-the-Woods College                           1369
## Saint Michael's College                                   1156
## Saint Olaf College                                         961
## Saint Peter's College                                      484
## Saint Vincent College                                      961
## Saint Xavier University                                    225
## Salem-Teikyo University                                     81
## Salem College                                             2116
## Salisbury State University                                 324
## Samford University                                         289
## San Diego State University                                  49
## Santa Clara University                                     361
## Sarah Lawrence College                                     324
## Savannah Coll. of Art and Design                           676
## Schreiner College                                          529
## Scripps College                                           1681
## Seattle Pacific University                                 400
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## Seton Hall University                                      225
## Seton Hill College                                        1369
## Shippensburg University of Penn.                           169
## Shorter College                                            324
## Siena College                                             1764
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## Simmons College                                           1089
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## Sioux Falls College                                         49
## Skidmore College                                           841
## Smith College                                             1936
## South Dakota State University                              841
## Southeast Missouri State University                         64
## Southeastern Oklahoma State Univ.                           81
## Southern California College                                121
## Southern Illinois University at Edwardsville                64
## Southern Methodist University                              289
## Southwest Baptist University                               169
## Southwest Missouri State University                        121
## Southwest State University                                 961
## Southwestern Adventist College                             144
## Southwestern College                                       144
## Southwestern University                                   1225
## Spalding University                                       1600
## Spelman College                                            324
## Spring Arbor College                                        81
## St. Bonaventure University                                1024
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## St. Martin's College                                        64
## St. Mary's College of California                           289
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## St. Mary's University of San Antonio                        49
## St. Norbert College                                       1296
## St. Paul's College                                          64
## St. Thomas Aquinas College                                 169
## Stephens College                                           289
## Stetson University                                         576
## Stevens Institute of Technology                           1089
## Stockton College of New Jersey                              49
## Stonehill College                                          900
## SUNY at Albany                                             256
## SUNY at Binghamton                                         225
## SUNY at Buffalo                                            225
## SUNY at Stony Brook                                         49
## SUNY College  at Brockport                                 196
## SUNY College  at Oswego                                    441
## SUNY College at Buffalo                                    144
## SUNY College at Cortland                                   289
## SUNY College at Fredonia                                   100
## SUNY College at Geneseo                                    625
## SUNY College at New Paltz                                   64
## SUNY College at Plattsburgh                                256
## SUNY College at Potsdam                                    289
## SUNY College at Purchase                                    64
## Susquehanna University                                    1369
## Sweet Briar College                                       2304
## Syracuse University                                        169
## Tabor College                                              225
## Talladega College                                           49
## Taylor University                                         1024
## Tennessee Wesleyan College                                 256
## Texas A&M Univ. at College Station                         841
## Texas A&M University at Galveston                          256
## Texas Christian University                                 529
## Texas Lutheran College                                     576
## Texas Southern University                                  441
## Texas Wesleyan University                                  100
## The Citadel                                                289
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## Tiffin University                                         1600
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## Trenton State College                                       36
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## Trinity College CT                                        2304
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## Trinity College VT                                         676
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## Tusculum College                                           784
## Tuskegee University                                         49
## Union College KY                                            81
## Union College NY                                          2401
## Univ. of Wisconsin at OshKosh                              196
## University of Alabama at Birmingham                        256
## University of Arkansas at Fayetteville                     100
## University of California at Berkeley                       100
## University of California at Irvine                         121
## University of Central Florida                               81
## University of Charleston                                   100
## University of Chicago                                     1296
## University of Cincinnati                                    36
## University of Connecticut at Storrs                        256
## University of Dallas                                       676
## University of Dayton                                       625
## University of Delaware                                     225
## University of Denver                                       441
## University of Detroit Mercy                                196
## University of Dubuque                                      324
## University of Evansville                                   676
## University of Florida                                      400
## University of Georgia                                      484
## University of Hartford                                      81
## University of Hawaii at Manoa                               36
## University of Illinois - Urbana                            169
## University of Illinois at Chicago                           36
## University of Indianapolis                                 529
## University of Kansas                                       289
## University of La Verne                                     529
## University of Louisville                                   576
## University of Maine at Farmington                          676
## University of Maine at Machias                              16
## University of Maine at Presque Isle                        121
## University of Maryland at Baltimore County                  36
## University of Maryland at College Park                     144
## University of Massachusetts at Amherst                     225
## University of Massachusetts at Dartmouth                   400
## University of Miami                                        289
## University of Michigan at Ann Arbor                        676
## University of Minnesota at Duluth                          121
## University of Minnesota at Morris                          256
## University of Minnesota Twin Cities                       1369
## University of Mississippi                                  196
## University of Missouri at Columbia                         225
## University of Missouri at Rolla                            529
## University of Missouri at Saint Louis                      225
## University of Mobile                                        16
## University of Montevallo                                    64
## University of Nebraska at Lincoln                         2304
## University of New England                                  169
## University of New Hampshire                                256
## University of North Carolina at Asheville                  121
## University of North Carolina at Chapel Hill                529
## University of North Carolina at Charlotte                   49
## University of North Carolina at Greensboro                 289
## University of North Carolina at Wilmington                 225
## University of North Dakota                                 256
## University of North Florida                                196
## University of North Texas                                   36
## University of Northern Colorado                             64
## University of Northern Iowa                                676
## University of Notre Dame                                  2116
## University of Oklahoma                                     121
## University of Oregon                                       169
## University of Pennsylvania                                1444
## University of Pittsburgh-Main Campus                       100
## University of Portland                                     289
## University of Puget Sound                                  289
## University of Rhode Island                                  49
## University of Richmond                                    1024
## University of Rochester                                    529
## University of San Diego                                    169
## University of San Francisco                                 64
## University of Sci. and Arts of Oklahoma                      9
## University of Scranton                                    1681
## University of South Carolina at Aiken                        9
## University of South Carolina at Columbia                   324
## University of South Florida                                 49
## University of Southern California                          100
## University of Southern Colorado                              0
## University of Southern Indiana                             441
## University of Southern Mississippi                         529
## University of St. Thomas MN                                169
## University of St. Thomas TX                                289
## University of Tennessee at Knoxville                       484
## University of Texas at Arlington                            16
## University of Texas at Austin                              121
## University of Texas at San Antonio                           9
## University of the Arts                                      81
## University of the Pacific                                  196
## University of the South                                   2704
## University of Tulsa                                        100
## University of Utah                                          81
## University of Vermont                                      100
## University of Virginia                                     484
## University of Washington                                   100
## University of West Florida                                 144
## University of Wisconsin-Stout                              289
## University of Wisconsin-Superior                           225
## University of Wisconsin-Whitewater                         256
## University of Wisconsin at Green Bay                         1
## University of Wisconsin at Madison                         400
## University of Wisconsin at Milwaukee                        64
## University of Wyoming                                      169
## Upper Iowa University                                      361
## Ursinus College                                           1600
## Ursuline College                                           225
## Valley City State University                               625
## Valparaiso University                                      529
## Vanderbilt University                                      676
## Vassar College                                            1521
## Villanova University                                       576
## Virginia Commonwealth University                           121
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## Virginia Tech                                              400
## Virginia Union University                                   64
## Virginia Wesleyan College                                  196
## Viterbo College                                            961
## Voorhees College                                             9
## Wabash College                                            3025
## Wagner College                                             529
## Wake Forest University                                    1369
## Walsh University                                          1089
## Warren Wilson College                                      400
## Wartburg College                                          1369
## Washington and Jefferson College                          1600
## Washington and Lee University                             2025
## Washington College                                        1369
## Washington State University                                900
## Washington University                                      961
## Wayne State College                                        841
## Waynesburg College                                         676
## Webber College                                              16
## Webster University                                         196
## Wellesley College                                         2601
## Wells College                                             1764
## Wentworth Institute of Technology                           64
## Wesley College                                             225
## Wesleyan University                                       1521
## West Chester University of Penn.                           256
## West Liberty State College                                 100
## West Virginia Wesleyan College                            1764
## Western Carolina University                                 81
## Western Maryland College                                  1521
## Western Michigan University                                121
## Western New England College                                225
## Western State College of Colorado                           16
## Western Washington University                              100
## Westfield State College                                    400
## Westminster College MO                                     400
## Westminster College                                       1681
## Westminster College of Salt Lake City                     1156
## Westmont College                                           289
## Wheaton College IL                                        1600
## Westminster College PA                                    1681
## Wheeling Jesuit College                                    729
## Whitman College                                           2601
## Whittier College                                           841
## Whitworth College                                          400
## Widener University                                         361
## Wilkes University                                          576
## Willamette University                                     1369
## William Jewell College                                     361
## William Woods University                                   256
## Williams College                                          4096
## Wilson College                                            1849
## Wingate College                                             64
## Winona State University                                    324
## Winthrop University                                        676
## Wisconsin Lutheran College                                 676
## Wittenberg University                                      841
## Wofford College                                           1764
## Worcester Polytechnic Institute                           1156
## Worcester State College                                    196
## Xavier University                                          961
## Xavier University of Louisiana                             400
## Yale University                                           2401
## York College of Pennsylvania                               784
## attr(,"assign")
## [1] 0 1 2
vcov(ajuste2) ### Matriz de variâncias e covariâncias dos estimadores.
##                   (Intercept)   perc.alumni I(perc.alumni^2)
## (Intercept)       3.526090777 -0.2679031981     4.254628e-03
## perc.alumni      -0.267903198  0.0242157266    -4.217689e-04
## I(perc.alumni^2)  0.004254628 -0.0004217689     7.959571e-06
### Vamos visualizar o ajuste do modelo
ggplot(College, aes(x = perc.alumni, y = Grad.Rate)) + geom_point() +
    stat_smooth(method = "lm", formula = y ~ x + I(x^2)) +
    theme_bw(base_size = 14)

### Extraindo os intervalos de confiança (95%) para os parâmetros
confint(ajuste2)
##                       2.5 %       97.5 %
## (Intercept)      42.2100816 49.582406847
## perc.alumni       0.7804723  1.391423440
## I(perc.alumni^2) -0.0131900 -0.002113503

Intervalo de confiança para a resposta média e para uma predição

Considere interesse em estimar a resposta média em um ponto \(x′_0 = (1, x_{01], x_{02}, ..., x_{0k})\), ou seja, \(E (y |x_0)\).

A estimativa pontual é dada pelo valor ajustado pelo modelo em \(x_0\): \[ \widehat{E(y|x_0)} = \hat{y_0} = x′_0 \hat{\beta} \] O estimador apresentado é não viciado para a real resposta média, com variância:

\[ Var(\widehat{E(y|x_0)}) = x'_0 Var(\hat{\beta})x_0 \] [73]

Um intervalo de confiança 100(1 − α)% para a resposta média em x′ 0 = (1, x01, x02, …, x0k ) é dado por: (y |x0) ± tn−p,α/2 × √ x′ 0 ( ˆβ)x0. Considere agora que se deseja predizer a resposta em um ponto (novo indivíduo) x′ 0 = (1, x01, x02, …, x0k ). A estimativa pontual, novamente, é dada pelo valor ajustado de y em x′ 0: ˆy0 = x′ 0 ˆβ.

[74]

Neste caso, a variância de ˆy0 fica dada por: Var (ˆy0) = σ2 + x′ 0Var ( ˆβ)x0. Um intervalo de confiança 100(1 − α)% para a predição de uma nova observação em x0 fica dada por: ˆy0 ± tn−p,α/2 × √ ˆσ2 + x′ 0 ( ˆβ)x0, em que ˆσ2 = QMRes .

### vamos realizar algumas predições. Considere faculdades com os seguintes percentuais de ex-alunos contribuintes: 13, 28 e 45

new_data <- data.frame(perc.alumni = c(13,28,45))
predict(ajuste2, newdata = new_data, se.fit = TRUE)
## $fit
##        1        2        3 
## 58.72042 70.30381 79.26910 
## 
## $se.fit
##         1         2         3 
## 0.6820636 0.7449543 1.2061247 
## 
## $df
## [1] 774
## 
## $residual.scale
## [1] 14.91414
### Estimativas pontuais e erros padrões (nota: erros padrões para a resposta média)
predict(ajuste2, newdata = new_data, interval = 'confidence') # para grupo com essa média
##        fit      lwr      upr
## 1 58.72042 57.38151 60.05933
## 2 70.30381 68.84144 71.76618
## 3 79.26910 76.90143 81.63676
### fit = estimando os formados a partir de um número de doadores (13,28,45). lwr e upr = limites do 95% de confiaça.

### Estimativas pontiais e intervalos de confiança (95%) para a resposta **média**.
predict(ajuste2, newdata = new_data, interval = 'prediction') ## Para caso isolado (observação única)
##        fit      lwr       upr
## 1 58.72042 29.41286  88.02798
## 2 70.30381 40.99035  99.61727
## 3 79.26910 49.89656 108.64164
### fit = estimando os formados a partir de um número de doadores (13,28,45). lwr e upr = limites do 95% de confiaça

### Estimativas pontiais e intervalos de confiança (95%) para a predição de **uma nova observação**.
### Vamos plotar as bandas de confiança e predição. Para isso, vamos preparar uma base de dados com os valores ajustados e os ICs(95%) para a resposta média e para as predições.

pred_int <- predict(ajuste2, interval="prediction")
med_int <- predict(ajuste2, interval="confidence")
data_pred <- data.frame(pred_lwr = pred_int[,'lwr'], pred_upr = pred_int[,'upr'],
                        med_lwr = med_int[,'lwr'], med_upr = med_int[,'upr'],
                        fit = med_int[,'fit'], perc.alumni = College$perc.alumni,
                        Grad.Rate = College$Grad.Rate)

ggplot(data_pred, aes(x = perc.alumni, y = Grad.Rate))+
    geom_point() +
    geom_line(aes(y=med_lwr), color = "red", linetype = "dashed", linewidth = 1.25) +
    geom_line(aes(y=med_upr), color = "red", linetype = "dashed", linewidth = 1.25) +
    geom_line(aes(y=pred_lwr), color = "green", linetype = "dashed", linewidth = 1.25) +
    geom_line(aes(y=pred_upr), color = "green", linetype = "dashed", linewidth = 1.25) +
    geom_line(aes(y=fit), color = "black", linetype = "dashed", linewidth = 1.25) +
    theme_bw(base_size = 14)
### Diagnóstico do ajuste (análise de resíduos)

### Vamos produzir alguns gráficos para os resíduos

fit_aj2 <- fitted(ajuste2) ### Vetor de valores ajustados
resid_aj2 <- rstandard(ajuste2) ### Vetor de resíduos padronizados

data_fit <- data.frame(y = College$Grad.Rate, fit_aj2, resid_aj2)

ggplot(data_fit, aes(x=y, y=fit_aj2)) + geom_point() + stat_smooth(method="lm") +
    theme_bw(base_size = 14)
### Gráfico de valores observados versus valores ajustados. 

ggplot(ajuste2, aes(x=fit_aj2, y=resid_aj2)) + geom_point() +
    stat_smooth(method="loess") + geom_hline(yintercept=0, col="red", linetype="dashed") +
    theme_bw(base_size = 14) +
    xlab('Valores ajustados') +
    ylab('Resíduos')
### Gráfico de resíduos versus valores ajustados

qqPlot(ajuste2)
### Gráfico quantil-quantil para os resíduos
### Parte 2 - Regressão linear múltipla, incluindo todas as variáveis da base como explicativas (exceto a taxa de formação, que é a resposta)

ajuste_p2 <- lm(Grad.Rate ~., data = College)
### Ajuste da regressão linear múltipla. A especificação "~." indica que
### todas as demais variáveis da base devem ser incluídas como explicativas.
### A título de ilustração, se quiséssemos ajustar um modelo apenas com as
### variáveis "Private", "Apps" e "Accept":
summary(ajuste_p2) 
## 
## Call:
## lm(formula = Grad.Rate ~ ., data = College)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -53.897  -7.132  -0.292   7.213  54.056 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 33.8736716  4.8480858   6.987 6.15e-12 ***
## PrivateYes   3.3813758  1.6965147   1.993 0.046605 *  
## Apps         0.0012984  0.0004418   2.939 0.003390 ** 
## Accept      -0.0006961  0.0008627  -0.807 0.419995    
## Enroll       0.0021593  0.0023081   0.936 0.349814    
## Top10perc    0.0548964  0.0717587   0.765 0.444501    
## Top25perc    0.1351288  0.0549667   2.458 0.014179 *  
## F.Undergrad -0.0004712  0.0004008  -1.176 0.240138    
## P.Undergrad -0.0014836  0.0003902  -3.802 0.000155 ***
## Outstate     0.0010174  0.0002334   4.359 1.49e-05 ***
## Room.Board   0.0019143  0.0005908   3.240 0.001246 ** 
## Books       -0.0022205  0.0029168  -0.761 0.446739    
## Personal    -0.0016635  0.0007698  -2.161 0.031000 *  
## PhD          0.0872827  0.0568102   1.536 0.124859    
## Terminal    -0.0747023  0.0623172  -1.199 0.231002    
## S.F.Ratio    0.0758222  0.1593102   0.476 0.634254    
## perc.alumni  0.2793343  0.0491750   5.680 1.91e-08 ***
## Expend      -0.0004565  0.0001542  -2.961 0.003163 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 12.75 on 759 degrees of freedom
## Multiple R-squared:  0.4615, Adjusted R-squared:  0.4495 
## F-statistic: 38.27 on 17 and 759 DF,  p-value: < 2.2e-16
ajuste_p2 <- lm(Grad.Rate ~.-Top25perc, data = College)

## Todas menos a top 25. pq o top 10 e top 25 "se sobrepoe"
## "ajustado os efeitos das outras variaveis"

summary(ajuste_p2) 
## 
## Call:
## lm(formula = Grad.Rate ~ . - Top25perc, data = College)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -47.786  -7.120  -0.257   7.171  54.050 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 36.3495104  4.7580380   7.640 6.57e-14 ***
## PrivateYes   3.3824039  1.7021347   1.987 0.047264 *  
## Apps         0.0011831  0.0004407   2.685 0.007421 ** 
## Accept      -0.0003936  0.0008568  -0.459 0.646059    
## Enroll       0.0015802  0.0023036   0.686 0.492938    
## Top10perc    0.1951766  0.0436553   4.471 8.98e-06 ***
## F.Undergrad -0.0003934  0.0004009  -0.981 0.326837    
## P.Undergrad -0.0014830  0.0003915  -3.788 0.000164 ***
## Outstate     0.0010161  0.0002342   4.339 1.63e-05 ***
## Room.Board   0.0019288  0.0005927   3.254 0.001188 ** 
## Books       -0.0020595  0.0029258  -0.704 0.481692    
## Personal    -0.0016865  0.0007723  -2.184 0.029273 *  
## PhD          0.0876416  0.0569982   1.538 0.124558    
## Terminal    -0.0565580  0.0620836  -0.911 0.362585    
## S.F.Ratio    0.0725205  0.1598322   0.454 0.650153    
## perc.alumni  0.2896423  0.0491582   5.892 5.73e-09 ***
## Expend      -0.0005259  0.0001521  -3.459 0.000573 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 12.79 on 760 degrees of freedom
## Multiple R-squared:  0.4573, Adjusted R-squared:  0.4458 
## F-statistic: 40.02 on 16 and 760 DF,  p-value: < 2.2e-16

Exemplo de numero de comodos e area em preço do imovel. Em sepada os dois são significantes. No modelo tudo junto ajutado a area é positiva e os comodos são negativos (mais comodos numa mesma area, ou seja comodos menores, menos preço).

ajuste_p2 <- lm(Grad.Rate ~ Expend, data = College)

## Todas menos a top 25. pq o top 10 e top 25 "se sobrepoe"
## "ajustado os efeitos das outras variaveis"

summary(ajuste_p2) 
## 
## Call:
## lm(formula = Grad.Rate ~ Expend, data = College)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -60.179 -10.275   0.238  10.377  55.058 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 5.306e+01  1.194e+00   44.42   <2e-16 ***
## Expend      1.284e-03  1.088e-04   11.80   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 15.83 on 775 degrees of freedom
## Multiple R-squared:  0.1524, Adjusted R-squared:  0.1513 
## F-statistic: 139.3 on 1 and 775 DF,  p-value: < 2.2e-16
ajuste_p2_ilustrativo <- lm(Grad.Rate ~ Private + Apps + Accept, data = College)

summary(ajuste_p2_ilustrativo) 
## 
## Call:
## lm(formula = Grad.Rate ~ Private + Apps + Accept, data = College)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -51.513  -8.856   0.228  10.516  50.053 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 48.5068491  1.4308933  33.900  < 2e-16 ***
## PrivateYes  17.7697914  1.3831646  12.847  < 2e-16 ***
## Apps         0.0030589  0.0004228   7.235 1.12e-12 ***
## Accept      -0.0025493  0.0006842  -3.726 0.000209 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 15.09 on 773 degrees of freedom
## Multiple R-squared:  0.2316, Adjusted R-squared:  0.2287 
## F-statistic: 77.68 on 3 and 773 DF,  p-value: < 2.2e-16
### Resumo do ajuste. Observe as variáveis com efeito significativo e os respectivos sinais.
### Variáveis com p-valor (Pr(>|t|)) < 0.05 podem ser consideradas com efeito 
### significativo na taxa de formados. Desta forma, as variáveis com efeito
### significativo na taxa de formados são Private (Yes), Apps, Top25perc, Outstate,
### Room.Board, 0.2793343; Já as variáveis com efeito negativo na taxa de formados
### são P.Undergrad, Personal e Expend.
### Apenas para fins de discussão, vamos ajustar um modelo de regressão tendo como
### única variável explicativa o gasto por aluno.
ajuste_temp <- lm(Grad.Rate ~ Expend, data = College)
summary(ajuste_temp)

### Compare o efeito do gasto por aluno na taxa de formação produzida pelo
### modelo em que ajustamos também os efeitos das demais variáveis com o
### efeito produzido pelo modelo em que as demais variáveis não são consideradas.
### Qual a diferença? Como você a justifica?
par(mfrow = c(2,2))
plot(ajuste_p2, which = 1:4)
### Gráficos de resíduos.
### O gráfico do canto superior esquerdo indica que a variância dos resíduos varia
### um pouco com a média (fitted values), e que os resíduos apresentam algum desvio
### da normalidade (conforme o qqplot). Ainda, conforme o gráfico do canto inferior
### direito, algumas observações podem ser identificadas como possivelmente influentes,
### produzindo maiores valores para a distância de Cook.

### Embora tenhamos alguns indicativos (ainda que não tão severos) de falta de ajuste
### da regressão linear, para fins didáticos vamos seguir a análise com esse tipo
### de modelagem.
### Agora, vamos para a etapa de seleção de covariáveis. Para isso, vamos
### usar os recursos do pacote leaps. Vamos consultar a documentação da função
### regsubsets.

help("regsubsets")

all_reg <- regsubsets(Grad.Rate ~ ., method = "exhaustive", nvmax = 18, data = College)
### Explorando TODAS as regressões possíveis
all_reg
plot(all_reg, scale="r2") ### Resultados baseados no R2
plot(all_reg, scale="adjr2") ### Resultados baseados no R2 ajustado
plot(all_reg, scale="bic") ### Resultados baseados no BIC
### Nesses gráficos avaliamos os resultados dos critérios para os melhores
### modelos ajustados com cada número de covariáveis.

s1 <- summary(all_reg, matrix.logical=TRUE)
s1 ### A matriz lógica permite identificar as covariáveis selecionadas 
### em cada modelo.

s1$rsq ### Valores de R2 para cada um dos modelos selecionados
s1$adjr2 ### Valores de R2 ajustado para cada um dos modelos selecionados
s1$bic ### Valores de BIC para cada um dos modelos selecionados

which.max(s1$adjr2)
coef(all_reg, id = 12)
### O modelo com doze covariáveis produziu maior valor de R2 ajustado.

which.min(s1$bic) 
coef(all_reg, id = 7)
### O modelo com sete covariáveis produziu menor valor de BIC.

which.max(s1$rsq) 
coef(all_reg, id = 17)
### O modelo com 17 covariáveis produziu menor valor de R2 (obviamente).

### Vamos produzir alguns gráficos usando os resultados desta análise.
n_cov <- 1:17

plot(n_cov, s1$bic, type = 'b', xlab = 'Número de covariáveis', 
     ylab = 'BIC', las = 1, pch = 20)
axis(1,1:17)

plot(n_cov, s1$adjr2, type = 'b', xlab = 'Número de covariáveis', 
     ylab = 'Adjusted R2', las = 1, pch = 20)
axis(1,1:17)

plot(n_cov, s1$rsq, type = 'b', xlab = 'Número de covariáveis', 
     ylab = 'R2', las = 1, pch = 20)
axis(1,1:17)




## Seleção de variáveis explicativas

Princípio de Occam: Dentre as várias explicações possíveis para um fenômeno, a mais simples é a melhor

Fuechsel, técnico da IBM: Garbage in, garbage out


- Neste módulo vamos tratar da seleção de covariáveis para o ajuste de modelos de regressão
linear.
- O objetivo é identificar um modelo parcimonioso, capaz de proporcionar bom ajuste com a menor quantidade possível de parâmetros.
- Diferentes métodos podem ser aplicados na seleção de um subconjunto “ótimo” de
variáveis.
- Importante ter em mente que diferentes métodos de seleção, frequentemente, remetem a
modelos distintos (lembre-se: “All models are wrong but some are useful”)


### Por que não incluir todas as covariáveis no modelo?

1 Um dos objetivos principais da análise de regressão é explicar a relação entre as variáveis de maneira simples e interpretável;
2 Quanto maior o número de parâmetros no modelo, menos graus de liberdade para os resíduos, menor precisão para as inferências;
3 Quanto maior o número de variáveis incluídas no modelo, maior a possibilidade de multicolinearidade;
4 Quanto mais complexo (parametrizado) o modelo, melhor o ajuste da amostra, mas menor seu poder de generalização (baixo poder preditivo)


### Como proceder a seleção do modelo?

Antes de aplicar qualquer método analítico para seleção de covariáveis, é conveniente fazer uma pré-triagem de variáveis, buscando eliminar variáveis que, a título de exemplo:
- Sejam redundantes;
- Apresentem elevado erro de medida;
- Não estejam no contexto do estudo;
- Apresentem elevada taxa de dados missing. . .

### Critérios para avaliação e comparação de modelos

- No processo de seleção de covariáveis, diferentes critérios podem ser usados para comparar os modelos produzidos. Alguns deles são descritos na sequência.
- **Coeficiente de deteminação** ($R^2$) - O coeficiente de determinação corresponde à proporção da variação dos dados explicada pela regressão:

$$
R^2 = \frac{SQ_{Total}-SQ_{R}^{es}}{SQ_{total}} = 1 = \frac{SQ_{Res}}{SQ_{total}}
$$

em que

$$
SQ_{Total} = \sum^n_{i-1}{(y_i - \overline{y})^2} \space \space e \space \space SQ_{Res} = \sum^n_{i-1}{(y_i - \hat{y}_i)^2}
$$

são as somas de quadrados total e atribuída aos resíduos, respectivamente
summary(ajuste22)
## 
## Call:
## lm(formula = Grad.Rate ~ ., data = College)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -53.897  -7.132  -0.292   7.213  54.056 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 33.8736716  4.8480858   6.987 6.15e-12 ***
## PrivateYes   3.3813758  1.6965147   1.993 0.046605 *  
## Apps         0.0012984  0.0004418   2.939 0.003390 ** 
## Accept      -0.0006961  0.0008627  -0.807 0.419995    
## Enroll       0.0021593  0.0023081   0.936 0.349814    
## Top10perc    0.0548964  0.0717587   0.765 0.444501    
## Top25perc    0.1351288  0.0549667   2.458 0.014179 *  
## F.Undergrad -0.0004712  0.0004008  -1.176 0.240138    
## P.Undergrad -0.0014836  0.0003902  -3.802 0.000155 ***
## Outstate     0.0010174  0.0002334   4.359 1.49e-05 ***
## Room.Board   0.0019143  0.0005908   3.240 0.001246 ** 
## Books       -0.0022205  0.0029168  -0.761 0.446739    
## Personal    -0.0016635  0.0007698  -2.161 0.031000 *  
## PhD          0.0872827  0.0568102   1.536 0.124859    
## Terminal    -0.0747023  0.0623172  -1.199 0.231002    
## S.F.Ratio    0.0758222  0.1593102   0.476 0.634254    
## perc.alumni  0.2793343  0.0491750   5.680 1.91e-08 ***
## Expend      -0.0004565  0.0001542  -2.961 0.003163 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 12.75 on 759 degrees of freedom
## Multiple R-squared:  0.4615, Adjusted R-squared:  0.4495 
## F-statistic: 38.27 on 17 and 759 DF,  p-value: < 2.2e-16

Multiple R-squared: 0.4615

Critérios para avaliação e comparação de modelos

  • O coeficiente de determinação expressa a proporção da variabilidade total explicada pelo modelo ajustado.

  • O valor de \(R^2\) nunca decresce à medida que novas covariáveis são incluídas no modelo.

  • Assim, não se deve optar pela seleção do modelo que produz maior \(R^2\), pois esse modelo incluiria, necessariamente, o maior número possível de covariáveis.

  • Coeficiente de deteminação ajustado - O coeficiente de determinação ajustado (ou simplesmente \(R^2\) ajustado) é definido por:

\[ R_{Aj}^2 = 1 - (\frac{n-1}{n-p}) (1 - R^2) \]

em que n e p são o número de observações e o número de parâmetros do modelo.

  • Diferentemente do que ocorre para \(R_2\), o valor de \(R_{Aj}^2\) pode não aumentar mediante inclusão de novas variáveis ao modelo. Deve-se optar por modelos com maiores valores de \(R_{Aj}^2\).

  • Tem problemas com overfiting, pq o \(R^2\) nunca diminui (podendo aumentar) quando aumenta o numero de parametros (mais \(\beta\)s)

resumo22 <- summary(ajuste22)
resumo22$r.squared
## [1] 0.4615405

Critérios de Informação

Akaike Information Criterion (AIC)

  • O critério de informação de Akaike (Akaike Information Criterion), ou simplesmente AIC, é definido por: \[ AIC = -2l(\hat{\theta}) + 2p \]

em que \(l(\hat{\theta})\) é a log-verossimilhança maximizada do modelo (calculada com base nos emv’s dos parâmetros) e p o número de parâmetros do modelo (evitar overfiting).

  • Basicamente quero modelo que explique bem com poucos parametros.

  • O AIC pode ser usado para qualquer modelo ajustado por máxima verossimilhança. No caso de um modelo de regressão linear temos:

\[ AIC = −n \space ln(SQ_{Res} /n) + 2p. \]

  • O componente 2p, na expressão do AIC , atua como termo de penalização atribuído à complexidade (número de parâmetros) do modelo. Pelo dobro, no caso.
AIC(ajuste22)
## [1] 6180.007

Critério de Informação Bayesiano (BIC)

  • Um critério alternativo ao AIC é o Critério de Informação Bayesiano (BIC), definido, para um modelo de regressão linear, por:

\[ BIC = −n \space ln(SQ_{Res} /n) + ln(n)p \]

  • O BIC penaliza mais fortemente a complexidade (número de parâmetros) do modelo que o AIC ao substituir p por \(ln(n)\) como fator de penalização. O \(ln(n)\) (BIC) penaliza mais de o \(2p\) (AIC).

  • Devemos selecionar modelos com menores valores de AIC (ou BIC).

BIC(ajuste22)
## [1] 6268.461

Seleção de modelos baseada em critérios de qualidade de ajuste

  • Um primeiro algoritmo de seleção de modelos, baseados em critérios de qualidade de ajuste, é o método todas as regressões possíveis;
  • Suponha uma análise de regressão com k variáveis;
  • Uma primeira alternativa consiste em ajustar todos os possíveis modelos de regressão com \(j \le k\) covariáveis, para \(j = 0, 1, 2, ..., k\);
  • Para cada modelo de regressão ajustado calcula-se o valor do critério de seleção escolhido \((AIC, BIC \space ou \space R_{Aj}^2 . . . )\);
  • O modelo selecionado será quele que apresentar o valor ótimo para o critério adotado (maior \(R_{Aj}^2\), menor AIC,. . . ).

\[ y \sim x_1, x_2, x_3 \] \(y \sim 1\)

\(y \sim x_1\)

\(y \sim x_2\)

\(y \sim x_3\)

\(y \sim x_1 + x_2\)

\(y \sim x_1 + x_3\)

\(y \sim x_2 + x_3\)

\(y \sim x_1 + x_2 + x_3\)

Para 3 variáveis \(2^3 = 8\) modelos

Busca exaustiva gera \(2^k\) modelos (sendo \(k\) o número de variaveis)

  • O método baseado em todas as regressões possíveis torna-se inviável mesmo para um número moderado de covariáveis;

  • Para k variáveis o número de regressões possíveis é \(2^k\) . Para k = 30, por exemplo, teríamos \(1.073.741.824\) modelos possíveis!

  • Como alternativa ao método de todas as regressões possíveis podemos usar os algoritmos backward, forward ou stepwise substituindo o teste F por algum dos critérios apresentado

Seleção de covariáveis - Método backward

Backward selection (or backward elimination)

  1. Ajuste o modelo de regressão com todas as k covariáveis disponíveis; 2.Remova uma a uma as covariáveis do modelo e calcule o valor do critério de qualidade de ajuste adotado (digamos o AIC, para ilustração);
  2. Remova do modelo (em definitivo) a variável cuja exclusão resultar em menor AIC;
  3. Os passos 1 a 3 são repetidos para o novo modelo (com a remoção da covariável) e o processo continua até o momento em que nenhuma remoção produzir um ajuste com menor AIC

Seleção de covariáveis - Método forward

Forward selection

  1. Ajuste o modelo nulo (sem covariáveis);
  2. Adicione uma a uma as covariáveis ao modelo e calcule o valor do critério de qualidade de ajuste adotado (digamos o AIC, para ilustração);
  3. Inclua no modelo (em definitivo) a variável cuja exclusão resultar em menor AIC;
  4. Os passos 1 a 3 são repetidos para o novo modelo (com a inclusão da covariável) e o processo continua até o momento em que nenhuma inclusão produzir um ajuste com menor AIC

Seleção de covariáveis - Método combinado

Stepwise selection (or sequential replacement)

  1. Ajuste o modelo de regressão com todas as k covariáveis disponíveis. Proceda como descrito no algoritmo backward, mas. . .
  2. A cada passo do algoritmo, avalie uma a uma tanto a inclusão das variáveis não inseridas quanto a exclusão das variáveis que compõem o modelo.
  3. Inclua (exclua) no modelo a variável cuja inclusão (exclusão) resultar em menor AIC;
  4. Os passos 1 a 3 são repetidos para o novo modelo (com a inclusão (exclusão) da covariável) e o processo continua até o momento em que nenhuma inclusão (ou exclusão) produzir um ajuste com menor AIC.
### Para finalizar, vamos aplicar os algoritmos de seleção do tipo
### stepwise. Primeiramente fixando k = 2, estamos definindo o AIC como critério de
### seleção, temos:

aj_full <- ajuste22

### Método backward
step_back_AIC <- step(aj_full, direction = "backward", data = College, k = 2)
## Start:  AIC=3972.98
## Grad.Rate ~ Private + Apps + Accept + Enroll + Top10perc + Top25perc + 
##     F.Undergrad + P.Undergrad + Outstate + Room.Board + Books + 
##     Personal + PhD + Terminal + S.F.Ratio + perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - S.F.Ratio    1      36.8 123332 3971.2
## - Books        1      94.1 123389 3971.6
## - Top10perc    1      95.1 123390 3971.6
## - Accept       1     105.8 123401 3971.6
## - Enroll       1     142.2 123437 3971.9
## - F.Undergrad  1     224.5 123519 3972.4
## - Terminal     1     233.4 123528 3972.4
## <none>                     123295 3973.0
## - PhD          1     383.4 123678 3973.4
## - Private      1     645.3 123940 3975.0
## - Personal     1     758.7 124054 3975.7
## - Top25perc    1     981.7 124277 3977.1
## - Apps         1    1403.4 124698 3979.8
## - Expend       1    1424.2 124719 3979.9
## - Room.Board   1    1705.5 125000 3981.7
## - P.Undergrad  1    2348.7 125644 3985.6
## - Outstate     1    3086.3 126381 3990.2
## - perc.alumni  1    5241.6 128537 4003.3
## 
## Step:  AIC=3971.21
## Grad.Rate ~ Private + Apps + Accept + Enroll + Top10perc + Top25perc + 
##     F.Undergrad + P.Undergrad + Outstate + Room.Board + Books + 
##     Personal + PhD + Terminal + perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - Books        1      91.7 123423 3969.8
## - Top10perc    1      93.4 123425 3969.8
## - Accept       1     109.1 123441 3969.9
## - Enroll       1     141.2 123473 3970.1
## - F.Undergrad  1     216.4 123548 3970.6
## - Terminal     1     237.6 123569 3970.7
## <none>                     123332 3971.2
## - PhD          1     400.2 123732 3971.7
## - Private      1     613.0 123945 3973.1
## - Personal     1     788.4 124120 3974.2
## - Top25perc    1     978.6 124310 3975.3
## - Apps         1    1426.8 124759 3978.1
## - Room.Board   1    1705.0 125037 3979.9
## - Expend       1    1854.2 125186 3980.8
## - P.Undergrad  1    2356.0 125688 3983.9
## - Outstate     1    3055.0 126387 3988.2
## - perc.alumni  1    5207.4 128539 4001.3
## 
## Step:  AIC=3969.79
## Grad.Rate ~ Private + Apps + Accept + Enroll + Top10perc + Top25perc + 
##     F.Undergrad + P.Undergrad + Outstate + Room.Board + Personal + 
##     PhD + Terminal + perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - Top10perc    1      86.7 123510 3968.3
## - Accept       1     110.1 123533 3968.5
## - Enroll       1     140.9 123564 3968.7
## - F.Undergrad  1     218.1 123641 3969.2
## - Terminal     1     279.1 123703 3969.5
## <none>                     123423 3969.8
## - PhD          1     469.5 123893 3970.7
## - Private      1     599.2 124023 3971.5
## - Personal     1     908.6 124332 3973.5
## - Top25perc    1     965.8 124389 3973.8
## - Apps         1    1425.9 124849 3976.7
## - Room.Board   1    1638.9 125062 3978.0
## - Expend       1    1885.5 125309 3979.6
## - P.Undergrad  1    2365.4 125789 3982.5
## - Outstate     1    3107.2 126531 3987.1
## - perc.alumni  1    5293.4 128717 4000.4
## 
## Step:  AIC=3968.33
## Grad.Rate ~ Private + Apps + Accept + Enroll + Top25perc + F.Undergrad + 
##     P.Undergrad + Outstate + Room.Board + Personal + PhD + Terminal + 
##     perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - Enroll       1     181.1 123691 3967.5
## - Accept       1     208.0 123718 3967.6
## - F.Undergrad  1     227.5 123738 3967.8
## - Terminal     1     315.0 123825 3968.3
## <none>                     123510 3968.3
## - PhD          1     533.2 124043 3969.7
## - Private      1     628.0 124138 3970.3
## - Personal     1     902.6 124413 3972.0
## - Room.Board   1    1602.8 125113 3976.3
## - Expend       1    1816.3 125326 3977.7
## - Apps         1    1835.9 125346 3977.8
## - P.Undergrad  1    2491.0 126001 3981.8
## - Outstate     1    3241.9 126752 3986.5
## - Top25perc    1    4063.5 127574 3991.5
## - perc.alumni  1    5334.0 128844 3999.2
## 
## Step:  AIC=3967.47
## Grad.Rate ~ Private + Apps + Accept + Top25perc + F.Undergrad + 
##     P.Undergrad + Outstate + Room.Board + Personal + PhD + Terminal + 
##     perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - F.Undergrad  1      53.0 123744 3965.8
## - Accept       1      95.1 123786 3966.1
## <none>                     123691 3967.5
## - Terminal     1     338.0 124029 3967.6
## - PhD          1     545.3 124236 3968.9
## - Private      1     631.2 124322 3969.4
## - Personal     1     895.3 124586 3971.1
## - Room.Board   1    1523.1 125214 3975.0
## - Expend       1    1715.1 125406 3976.2
## - Apps         1    1720.1 125411 3976.2
## - P.Undergrad  1    2613.8 126305 3981.7
## - Outstate     1    3190.0 126881 3985.3
## - Top25perc    1    4126.3 127817 3991.0
## - perc.alumni  1    5621.6 129313 4000.0
## 
## Step:  AIC=3965.8
## Grad.Rate ~ Private + Apps + Accept + Top25perc + P.Undergrad + 
##     Outstate + Room.Board + Personal + PhD + Terminal + perc.alumni + 
##     Expend
## 
##               Df Sum of Sq    RSS    AIC
## - Accept       1     251.7 123996 3965.4
## <none>                     123744 3965.8
## - Terminal     1     350.2 124094 3966.0
## - PhD          1     549.6 124294 3967.2
## - Private      1     749.8 124494 3968.5
## - Personal     1     973.0 124717 3969.9
## - Room.Board   1    1572.3 125317 3973.6
## - Expend       1    1750.9 125495 3974.7
## - Apps         1    1775.8 125520 3974.9
## - P.Undergrad  1    3195.5 126940 3983.6
## - Outstate     1    3415.9 127160 3985.0
## - Top25perc    1    4094.9 127839 3989.1
## - perc.alumni  1    5579.5 129324 3998.1
## 
## Step:  AIC=3965.38
## Grad.Rate ~ Private + Apps + Top25perc + P.Undergrad + Outstate + 
##     Room.Board + Personal + PhD + Terminal + perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## <none>                     123996 3965.4
## - Terminal     1     391.0 124387 3965.8
## - PhD          1     524.0 124520 3966.7
## - Private      1     785.5 124781 3968.3
## - Personal     1     992.0 124988 3969.6
## - Expend       1    1512.5 125508 3972.8
## - Room.Board   1    1705.6 125701 3974.0
## - Outstate     1    3221.2 127217 3983.3
## - P.Undergrad  1    3449.0 127445 3984.7
## - Top25perc    1    4503.8 128500 3991.1
## - Apps         1    5016.2 129012 3994.2
## - perc.alumni  1    5748.0 129744 3998.6
summary(step_back_AIC)
## 
## Call:
## lm(formula = Grad.Rate ~ Private + Apps + Top25perc + P.Undergrad + 
##     Outstate + Room.Board + Personal + PhD + Terminal + perc.alumni + 
##     Expend, data = College)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -51.684  -7.488  -0.282   7.363  53.482 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 33.4888648  3.3489573  10.000  < 2e-16 ***
## PrivateYes   3.5847682  1.6283712   2.201  0.02800 *  
## Apps         0.0008950  0.0001609   5.563 3.67e-08 ***
## Top25perc    0.1697318  0.0321993   5.271 1.76e-07 ***
## P.Undergrad -0.0016749  0.0003631  -4.613 4.65e-06 ***
## Outstate     0.0010061  0.0002257   4.458 9.51e-06 ***
## Room.Board   0.0018799  0.0005795   3.244  0.00123 ** 
## Personal    -0.0018516  0.0007485  -2.474  0.01358 *  
## PhD          0.0997365  0.0554704   1.798  0.07257 .  
## Terminal    -0.0950484  0.0612000  -1.553  0.12082    
## perc.alumni  0.2887259  0.0484841   5.955 3.96e-09 ***
## Expend      -0.0003942  0.0001290  -3.055  0.00233 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 12.73 on 765 degrees of freedom
## Multiple R-squared:  0.4585, Adjusted R-squared:  0.4507 
## F-statistic: 58.88 on 11 and 765 DF,  p-value: < 2.2e-16
### Método forward. Para o método forward devemos definir o escopo da seleção
### (menor e maior modelo). O menor seria o modelo nulo (apenas com o intercepto),
### enquanto o maior seria o modelo com todas as covariáveis.
aj_lower <- lm(Grad.Rate~1, data = College)

aj_upper <- lm(Grad.Rate~., data = College)
formula(aj_upper)
## Grad.Rate ~ Private + Apps + Accept + Enroll + Top10perc + Top25perc + 
##     F.Undergrad + P.Undergrad + Outstate + Room.Board + Books + 
##     Personal + PhD + Terminal + S.F.Ratio + perc.alumni + Expend
step_for_AIC <- step(aj_lower, direction = "forward", scope=formula(aj_upper), 
                     data = College, k = 2)
## Start:  AIC=4419.97
## Grad.Rate ~ 1
## 
##               Df Sum of Sq    RSS    AIC
## + Outstate     1     74732 154245 4115.0
## + Top10perc    1     56103 172875 4203.6
## + perc.alumni  1     55179 173798 4207.7
## + Top25perc    1     52160 176817 4221.1
## + Room.Board   1     41348 187630 4267.2
## + Expend       1     34889 194089 4293.5
## + Private      1     25876 203102 4328.8
## + S.F.Ratio    1     21540 207437 4345.2
## + PhD          1     21306 207671 4346.1
## + Terminal     1     19194 209783 4353.9
## + Personal     1     16611 212366 4363.5
## + P.Undergrad  1     15124 213853 4368.9
## + Apps         1      4931 224046 4405.1
## + F.Undergrad  1      1421 227556 4417.1
## + Accept       1      1037 227940 4418.4
## <none>                     228977 4420.0
## + Enroll       1       114 228863 4421.6
## + Books        1         0 228977 4422.0
## 
## Step:  AIC=4115
## Grad.Rate ~ Outstate
## 
##               Df Sum of Sq    RSS    AIC
## + Top25perc    1   11767.7 142478 4055.3
## + Top10perc    1   10107.6 144138 4064.3
## + perc.alumni  1    9444.9 144800 4067.9
## + Apps         1    3201.7 151044 4100.7
## + P.Undergrad  1    3079.0 151166 4101.3
## + Personal     1    2438.8 151807 4104.6
## + PhD          1    1995.9 152250 4106.9
## + Accept       1    1541.6 152704 4109.2
## + Room.Board   1    1048.3 153197 4111.7
## + Enroll       1    1037.1 153208 4111.8
## + Terminal     1     875.4 153370 4112.6
## + F.Undergrad  1     475.1 153770 4114.6
## <none>                     154245 4115.0
## + Private      1     137.7 154108 4116.3
## + Books        1     102.5 154143 4116.5
## + S.F.Ratio    1      34.8 154211 4116.8
## + Expend       1      15.0 154230 4116.9
## 
## Step:  AIC=4055.34
## Grad.Rate ~ Outstate + Top25perc
## 
##               Df Sum of Sq    RSS    AIC
## + perc.alumni  1    5997.5 136480 4023.9
## + P.Undergrad  1    4196.4 138281 4034.1
## + Personal     1    3376.9 139101 4038.7
## + Private      1    1517.5 140960 4049.0
## + Expend       1     957.6 141520 4052.1
## + Room.Board   1     931.7 141546 4052.2
## + Books        1     496.1 141982 4054.6
## <none>                     142478 4055.3
## + F.Undergrad  1     334.0 142144 4055.5
## + Apps         1     292.3 142186 4055.7
## + Terminal     1     259.5 142218 4055.9
## + Top10perc    1     223.9 142254 4056.1
## + S.F.Ratio    1      87.5 142390 4056.9
## + Accept       1      52.3 142425 4057.1
## + Enroll       1      40.1 142438 4057.1
## + PhD          1      16.1 142462 4057.2
## 
## Step:  AIC=4023.92
## Grad.Rate ~ Outstate + Top25perc + perc.alumni
## 
##               Df Sum of Sq    RSS    AIC
## + P.Undergrad  1   2580.25 133900 4011.1
## + Personal     1   2096.79 134383 4013.9
## + Room.Board   1   1919.65 134561 4014.9
## + Apps         1   1347.54 135133 4018.2
## + Expend       1    958.10 135522 4020.4
## + Accept       1    758.79 135721 4021.6
## + Private      1    555.35 135925 4022.8
## + S.F.Ratio    1    370.29 136110 4023.8
## <none>                     136480 4023.9
## + Books        1    213.78 136266 4024.7
## + Terminal     1    178.28 136302 4024.9
## + Top10perc    1     97.21 136383 4025.4
## + Enroll       1     85.88 136394 4025.4
## + F.Undergrad  1      1.19 136479 4025.9
## + PhD          1      0.00 136480 4025.9
## 
## Step:  AIC=4011.09
## Grad.Rate ~ Outstate + Top25perc + perc.alumni + P.Undergrad
## 
##               Df Sum of Sq    RSS    AIC
## + Apps         1    3864.5 130035 3990.3
## + Accept       1    2892.9 131007 3996.1
## + Room.Board   1    2493.1 131407 3998.5
## + Enroll       1    1470.5 132430 4004.5
## + Personal     1    1213.4 132687 4006.0
## + F.Undergrad  1     953.7 132946 4007.5
## + Expend       1     668.5 133231 4009.2
## + S.F.Ratio    1     586.1 133314 4009.7
## <none>                     133900 4011.1
## + PhD          1     189.5 133711 4012.0
## + Books        1     125.1 133775 4012.4
## + Top10perc    1      64.3 133836 4012.7
## + Private      1      35.8 133864 4012.9
## + Terminal     1       0.0 133900 4013.1
## 
## Step:  AIC=3990.33
## Grad.Rate ~ Outstate + Top25perc + perc.alumni + P.Undergrad + 
##     Apps
## 
##               Df Sum of Sq    RSS    AIC
## + Room.Board   1   1862.58 128173 3981.1
## + Personal     1   1533.45 128502 3983.1
## + Expend       1   1519.13 128516 3983.2
## + Private      1   1163.67 128872 3985.4
## + F.Undergrad  1    736.15 129299 3987.9
## + Enroll       1    389.66 129646 3990.0
## <none>                     130035 3990.3
## + S.F.Ratio    1    282.41 129753 3990.6
## + Books        1    210.11 129825 3991.1
## + Terminal     1    134.41 129901 3991.5
## + Accept       1    107.62 129928 3991.7
## + Top10perc    1      4.46 130031 3992.3
## + PhD          1      2.19 130033 3992.3
## 
## Step:  AIC=3981.13
## Grad.Rate ~ Outstate + Top25perc + perc.alumni + P.Undergrad + 
##     Apps + Room.Board
## 
##               Df Sum of Sq    RSS    AIC
## + Expend       1   1814.50 126358 3972.0
## + Personal     1   1305.58 126867 3975.2
## + Private      1    890.33 127283 3977.7
## + F.Undergrad  1    465.62 127707 3980.3
## + Books        1    372.34 127801 3980.9
## + S.F.Ratio    1    353.01 127820 3981.0
## <none>                     128173 3981.1
## + Terminal     1    280.14 127893 3981.4
## + Enroll       1    165.70 128007 3982.1
## + Accept       1     34.78 128138 3982.9
## + PhD          1      1.56 128171 3983.1
## + Top10perc    1      1.43 128171 3983.1
## 
## Step:  AIC=3972.05
## Grad.Rate ~ Outstate + Top25perc + perc.alumni + P.Undergrad + 
##     Apps + Room.Board + Expend
## 
##               Df Sum of Sq    RSS    AIC
## + Personal     1   1013.08 125345 3967.8
## + Private      1    838.46 125520 3968.9
## + F.Undergrad  1    639.71 125719 3970.1
## + Accept       1    348.76 126010 3971.9
## <none>                     126358 3972.0
## + Top10perc    1    289.00 126069 3972.3
## + Books        1    275.25 126083 3972.4
## + Enroll       1    271.29 126087 3972.4
## + Terminal     1    189.86 126169 3972.9
## + PhD          1      4.21 126354 3974.0
## + S.F.Ratio    1      3.90 126354 3974.0
## 
## Step:  AIC=3967.79
## Grad.Rate ~ Outstate + Top25perc + perc.alumni + P.Undergrad + 
##     Apps + Room.Board + Expend + Personal
## 
##               Df Sum of Sq    RSS    AIC
## + Private      1    804.40 124541 3964.8
## + F.Undergrad  1    461.72 124884 3966.9
## + Accept       1    325.55 125020 3967.8
## <none>                     125345 3967.8
## + Top10perc    1    317.32 125028 3967.8
## + Terminal     1    188.21 125157 3968.6
## + Enroll       1    177.88 125167 3968.7
## + Books        1    128.70 125217 3969.0
## + PhD          1      5.38 125340 3969.8
## + S.F.Ratio    1      0.01 125345 3969.8
## 
## Step:  AIC=3964.79
## Grad.Rate ~ Outstate + Top25perc + perc.alumni + P.Undergrad + 
##     Apps + Room.Board + Expend + Personal + Private
## 
##               Df Sum of Sq    RSS    AIC
## <none>                     124541 3964.8
## + Top10perc    1   280.394 124261 3965.0
## + Accept       1   240.404 124301 3965.3
## + F.Undergrad  1   216.360 124325 3965.4
## + Books        1   159.065 124382 3965.8
## + PhD          1   154.100 124387 3965.8
## + Enroll       1    60.716 124480 3966.4
## + S.F.Ratio    1    37.680 124503 3966.6
## + Terminal     1    21.060 124520 3966.7
summary(step_for_AIC)
## 
## Call:
## lm(formula = Grad.Rate ~ Outstate + Top25perc + perc.alumni + 
##     P.Undergrad + Apps + Room.Board + Expend + Personal + Private, 
##     data = College)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -52.345  -7.551  -0.426   7.040  51.789 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 32.9174907  2.5635570  12.841  < 2e-16 ***
## Outstate     0.0010226  0.0002203   4.641 4.07e-06 ***
## Top25perc    0.1763996  0.0304582   5.792 1.02e-08 ***
## perc.alumni  0.2876422  0.0483114   5.954 3.98e-09 ***
## P.Undergrad -0.0016678  0.0003611  -4.619 4.52e-06 ***
## Apps         0.0009022  0.0001609   5.606 2.89e-08 ***
## Room.Board   0.0018262  0.0005732   3.186  0.00150 ** 
## Expend      -0.0003888  0.0001288  -3.019  0.00262 ** 
## Personal    -0.0018394  0.0007491  -2.455  0.01429 *  
## PrivateYes   3.3935160  1.5246563   2.226  0.02632 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 12.74 on 767 degrees of freedom
## Multiple R-squared:  0.4561, Adjusted R-squared:  0.4497 
## F-statistic: 71.46 on 9 and 767 DF,  p-value: < 2.2e-16
### Finalmente, o algoritmo que considera tanto exclusão quanto inclusão de
### covariáveis a cada passo
step_both_AIC <- step(aj_full, direction = "both", data = College, k = 2)
## Start:  AIC=3972.98
## Grad.Rate ~ Private + Apps + Accept + Enroll + Top10perc + Top25perc + 
##     F.Undergrad + P.Undergrad + Outstate + Room.Board + Books + 
##     Personal + PhD + Terminal + S.F.Ratio + perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - S.F.Ratio    1      36.8 123332 3971.2
## - Books        1      94.1 123389 3971.6
## - Top10perc    1      95.1 123390 3971.6
## - Accept       1     105.8 123401 3971.6
## - Enroll       1     142.2 123437 3971.9
## - F.Undergrad  1     224.5 123519 3972.4
## - Terminal     1     233.4 123528 3972.4
## <none>                     123295 3973.0
## - PhD          1     383.4 123678 3973.4
## - Private      1     645.3 123940 3975.0
## - Personal     1     758.7 124054 3975.7
## - Top25perc    1     981.7 124277 3977.1
## - Apps         1    1403.4 124698 3979.8
## - Expend       1    1424.2 124719 3979.9
## - Room.Board   1    1705.5 125000 3981.7
## - P.Undergrad  1    2348.7 125644 3985.6
## - Outstate     1    3086.3 126381 3990.2
## - perc.alumni  1    5241.6 128537 4003.3
## 
## Step:  AIC=3971.21
## Grad.Rate ~ Private + Apps + Accept + Enroll + Top10perc + Top25perc + 
##     F.Undergrad + P.Undergrad + Outstate + Room.Board + Books + 
##     Personal + PhD + Terminal + perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - Books        1      91.7 123423 3969.8
## - Top10perc    1      93.4 123425 3969.8
## - Accept       1     109.1 123441 3969.9
## - Enroll       1     141.2 123473 3970.1
## - F.Undergrad  1     216.4 123548 3970.6
## - Terminal     1     237.6 123569 3970.7
## <none>                     123332 3971.2
## - PhD          1     400.2 123732 3971.7
## + S.F.Ratio    1      36.8 123295 3973.0
## - Private      1     613.0 123945 3973.1
## - Personal     1     788.4 124120 3974.2
## - Top25perc    1     978.6 124310 3975.3
## - Apps         1    1426.8 124759 3978.1
## - Room.Board   1    1705.0 125037 3979.9
## - Expend       1    1854.2 125186 3980.8
## - P.Undergrad  1    2356.0 125688 3983.9
## - Outstate     1    3055.0 126387 3988.2
## - perc.alumni  1    5207.4 128539 4001.3
## 
## Step:  AIC=3969.79
## Grad.Rate ~ Private + Apps + Accept + Enroll + Top10perc + Top25perc + 
##     F.Undergrad + P.Undergrad + Outstate + Room.Board + Personal + 
##     PhD + Terminal + perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - Top10perc    1      86.7 123510 3968.3
## - Accept       1     110.1 123533 3968.5
## - Enroll       1     140.9 123564 3968.7
## - F.Undergrad  1     218.1 123641 3969.2
## - Terminal     1     279.1 123703 3969.5
## <none>                     123423 3969.8
## - PhD          1     469.5 123893 3970.7
## + Books        1      91.7 123332 3971.2
## - Private      1     599.2 124023 3971.5
## + S.F.Ratio    1      34.3 123389 3971.6
## - Personal     1     908.6 124332 3973.5
## - Top25perc    1     965.8 124389 3973.8
## - Apps         1    1425.9 124849 3976.7
## - Room.Board   1    1638.9 125062 3978.0
## - Expend       1    1885.5 125309 3979.6
## - P.Undergrad  1    2365.4 125789 3982.5
## - Outstate     1    3107.2 126531 3987.1
## - perc.alumni  1    5293.4 128717 4000.4
## 
## Step:  AIC=3968.33
## Grad.Rate ~ Private + Apps + Accept + Enroll + Top25perc + F.Undergrad + 
##     P.Undergrad + Outstate + Room.Board + Personal + PhD + Terminal + 
##     perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - Enroll       1     181.1 123691 3967.5
## - Accept       1     208.0 123718 3967.6
## - F.Undergrad  1     227.5 123738 3967.8
## - Terminal     1     315.0 123825 3968.3
## <none>                     123510 3968.3
## - PhD          1     533.2 124043 3969.7
## + Top10perc    1      86.7 123423 3969.8
## + Books        1      85.0 123425 3969.8
## + S.F.Ratio    1      32.9 123477 3970.1
## - Private      1     628.0 124138 3970.3
## - Personal     1     902.6 124413 3972.0
## - Room.Board   1    1602.8 125113 3976.3
## - Expend       1    1816.3 125326 3977.7
## - Apps         1    1835.9 125346 3977.8
## - P.Undergrad  1    2491.0 126001 3981.8
## - Outstate     1    3241.9 126752 3986.5
## - Top25perc    1    4063.5 127574 3991.5
## - perc.alumni  1    5334.0 128844 3999.2
## 
## Step:  AIC=3967.47
## Grad.Rate ~ Private + Apps + Accept + Top25perc + F.Undergrad + 
##     P.Undergrad + Outstate + Room.Board + Personal + PhD + Terminal + 
##     perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - F.Undergrad  1      53.0 123744 3965.8
## - Accept       1      95.1 123786 3966.1
## <none>                     123691 3967.5
## - Terminal     1     338.0 124029 3967.6
## + Enroll       1     181.1 123510 3968.3
## + Top10perc    1     126.9 123564 3968.7
## - PhD          1     545.3 124236 3968.9
## + Books        1      83.2 123608 3968.9
## + S.F.Ratio    1      31.5 123660 3969.3
## - Private      1     631.2 124322 3969.4
## - Personal     1     895.3 124586 3971.1
## - Room.Board   1    1523.1 125214 3975.0
## - Expend       1    1715.1 125406 3976.2
## - Apps         1    1720.1 125411 3976.2
## - P.Undergrad  1    2613.8 126305 3981.7
## - Outstate     1    3190.0 126881 3985.3
## - Top25perc    1    4126.3 127817 3991.0
## - perc.alumni  1    5621.6 129313 4000.0
## 
## Step:  AIC=3965.8
## Grad.Rate ~ Private + Apps + Accept + Top25perc + P.Undergrad + 
##     Outstate + Room.Board + Personal + PhD + Terminal + perc.alumni + 
##     Expend
## 
##               Df Sum of Sq    RSS    AIC
## - Accept       1     251.7 123996 3965.4
## <none>                     123744 3965.8
## - Terminal     1     350.2 124094 3966.0
## + Top10perc    1     102.4 123642 3967.2
## - PhD          1     549.6 124294 3967.2
## + Books        1      85.4 123659 3967.3
## + F.Undergrad  1      53.0 123691 3967.5
## + S.F.Ratio    1      26.2 123718 3967.6
## + Enroll       1       6.5 123738 3967.8
## - Private      1     749.8 124494 3968.5
## - Personal     1     973.0 124717 3969.9
## - Room.Board   1    1572.3 125317 3973.6
## - Expend       1    1750.9 125495 3974.7
## - Apps         1    1775.8 125520 3974.9
## - P.Undergrad  1    3195.5 126940 3983.6
## - Outstate     1    3415.9 127160 3985.0
## - Top25perc    1    4094.9 127839 3989.1
## - perc.alumni  1    5579.5 129324 3998.1
## 
## Step:  AIC=3965.38
## Grad.Rate ~ Private + Apps + Top25perc + P.Undergrad + Outstate + 
##     Room.Board + Personal + PhD + Terminal + perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## <none>                     123996 3965.4
## + Accept       1     251.7 123744 3965.8
## - Terminal     1     391.0 124387 3965.8
## + F.Undergrad  1     209.5 123786 3966.1
## + Top10perc    1     196.1 123800 3966.2
## - PhD          1     524.0 124520 3966.7
## + Books        1      86.5 123909 3966.8
## + Enroll       1      63.4 123932 3967.0
## + S.F.Ratio    1      25.0 123971 3967.2
## - Private      1     785.5 124781 3968.3
## - Personal     1     992.0 124988 3969.6
## - Expend       1    1512.5 125508 3972.8
## - Room.Board   1    1705.6 125701 3974.0
## - Outstate     1    3221.2 127217 3983.3
## - P.Undergrad  1    3449.0 127445 3984.7
## - Top25perc    1    4503.8 128500 3991.1
## - Apps         1    5016.2 129012 3994.2
## - perc.alumni  1    5748.0 129744 3998.6
summary(step_both_AIC)
## 
## Call:
## lm(formula = Grad.Rate ~ Private + Apps + Top25perc + P.Undergrad + 
##     Outstate + Room.Board + Personal + PhD + Terminal + perc.alumni + 
##     Expend, data = College)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -51.684  -7.488  -0.282   7.363  53.482 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 33.4888648  3.3489573  10.000  < 2e-16 ***
## PrivateYes   3.5847682  1.6283712   2.201  0.02800 *  
## Apps         0.0008950  0.0001609   5.563 3.67e-08 ***
## Top25perc    0.1697318  0.0321993   5.271 1.76e-07 ***
## P.Undergrad -0.0016749  0.0003631  -4.613 4.65e-06 ***
## Outstate     0.0010061  0.0002257   4.458 9.51e-06 ***
## Room.Board   0.0018799  0.0005795   3.244  0.00123 ** 
## Personal    -0.0018516  0.0007485  -2.474  0.01358 *  
## PhD          0.0997365  0.0554704   1.798  0.07257 .  
## Terminal    -0.0950484  0.0612000  -1.553  0.12082    
## perc.alumni  0.2887259  0.0484841   5.955 3.96e-09 ***
## Expend      -0.0003942  0.0001290  -3.055  0.00233 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 12.73 on 765 degrees of freedom
## Multiple R-squared:  0.4585, Adjusted R-squared:  0.4507 
## F-statistic: 58.88 on 11 and 765 DF,  p-value: < 2.2e-16
### Vamos comparar os ajustes.
data.frame(compareCoefs(step_back_AIC, step_for_AIC, step_both_AIC))
## Calls:
## 1: lm(formula = Grad.Rate ~ Private + Apps + Top25perc + P.Undergrad + 
##   Outstate + Room.Board + Personal + PhD + Terminal + perc.alumni + Expend, 
##   data = College)
## 2: lm(formula = Grad.Rate ~ Outstate + Top25perc + perc.alumni + 
##   P.Undergrad + Apps + Room.Board + Expend + Personal + Private, data = 
##   College)
## 3: lm(formula = Grad.Rate ~ Private + Apps + Top25perc + P.Undergrad + 
##   Outstate + Room.Board + Personal + PhD + Terminal + perc.alumni + Expend, 
##   data = College)
## 
##               Model 1   Model 2   Model 3
## (Intercept)     33.49     32.92     33.49
## SE               3.35      2.56      3.35
##                                          
## PrivateYes       3.58      3.39      3.58
## SE               1.63      1.52      1.63
##                                          
## Apps         0.000895  0.000902  0.000895
## SE           0.000161  0.000161  0.000161
##                                          
## Top25perc      0.1697    0.1764    0.1697
## SE             0.0322    0.0305    0.0322
##                                          
## P.Undergrad -0.001675 -0.001668 -0.001675
## SE           0.000363  0.000361  0.000363
##                                          
## Outstate     0.001006  0.001023  0.001006
## SE           0.000226  0.000220  0.000226
##                                          
## Room.Board   0.001880  0.001826  0.001880
## SE           0.000580  0.000573  0.000580
##                                          
## Personal    -0.001852 -0.001839 -0.001852
## SE           0.000748  0.000749  0.000748
##                                          
## PhD            0.0997              0.0997
## SE             0.0555              0.0555
##                                          
## Terminal      -0.0950             -0.0950
## SE             0.0612              0.0612
##                                          
## perc.alumni    0.2887    0.2876    0.2887
## SE             0.0485    0.0483    0.0485
##                                          
## Expend      -0.000394 -0.000389 -0.000394
## SE           0.000129  0.000129  0.000129
## 
##                    Model.1       Model.2       Model.3
## X.Intercept. 33.4888648432 32.9174907287 33.4888648432
## SE            3.3489572972  2.5635569714  3.3489572972
## X                       NA            NA            NA
## PrivateYes    3.5847682354  3.3935159568  3.5847682354
## SE.1          1.6283711996  1.5246563455  1.6283711996
## X.1                     NA            NA            NA
## Apps          0.0008949751  0.0009022273  0.0008949751
## SE.2          0.0001608786  0.0001609453  0.0001608786
## X.2                     NA            NA            NA
## Top25perc     0.1697317832  0.1763995851  0.1697317832
## SE.3          0.0321992964  0.0304582482  0.0321992964
## X.3                     NA            NA            NA
## P.Undergrad  -0.0016748544 -0.0016677982 -0.0016748544
## SE.4          0.0003630820  0.0003610642  0.0003630820
## X.4                     NA            NA            NA
## Outstate      0.0010060810  0.0010225578  0.0010060810
## SE.5          0.0002256830  0.0002203085  0.0002256830
## X.5                     NA            NA            NA
## Room.Board    0.0018799059  0.0018262352  0.0018799059
## SE.6          0.0005795294  0.0005732225  0.0005795294
## X.6                     NA            NA            NA
## Personal     -0.0018516261 -0.0018393858 -0.0018516261
## SE.7          0.0007484607  0.0007490924  0.0007484607
## X.7                     NA            NA            NA
## PhD           0.0997365432            NA  0.0997365432
## SE.8          0.0554704147            NA  0.0554704147
## X.8                     NA            NA            NA
## Terminal     -0.0950484084            NA -0.0950484084
## SE.9          0.0611999674            NA  0.0611999674
## X.9                     NA            NA            NA
## perc.alumni   0.2887259430  0.2876421551  0.2887259430
## SE.10         0.0484841126  0.0483114341  0.0484841126
## X.10                    NA            NA            NA
## Expend       -0.0003942095 -0.0003888320 -0.0003942095
## SE.11         0.0001290471  0.0001287827  0.0001290471
## X.11                    NA            NA            NA
### Agora fixando k = log(n) (critério BIC):

### Método backward
step_back_BIC <- step(aj_full, direction = "backward", data = College, k = log(nrow(College)))
## Start:  AIC=4056.77
## Grad.Rate ~ Private + Apps + Accept + Enroll + Top10perc + Top25perc + 
##     F.Undergrad + P.Undergrad + Outstate + Room.Board + Books + 
##     Personal + PhD + Terminal + S.F.Ratio + perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - S.F.Ratio    1      36.8 123332 4050.4
## - Books        1      94.1 123389 4050.7
## - Top10perc    1      95.1 123390 4050.7
## - Accept       1     105.8 123401 4050.8
## - Enroll       1     142.2 123437 4051.0
## - F.Undergrad  1     224.5 123519 4051.5
## - Terminal     1     233.4 123528 4051.6
## - PhD          1     383.4 123678 4052.5
## - Private      1     645.3 123940 4054.2
## - Personal     1     758.7 124054 4054.9
## - Top25perc    1     981.7 124277 4056.3
## <none>                     123295 4056.8
## - Apps         1    1403.4 124698 4058.9
## - Expend       1    1424.2 124719 4059.0
## - Room.Board   1    1705.5 125000 4060.8
## - P.Undergrad  1    2348.7 125644 4064.8
## - Outstate     1    3086.3 126381 4069.3
## - perc.alumni  1    5241.6 128537 4082.5
## 
## Step:  AIC=4050.35
## Grad.Rate ~ Private + Apps + Accept + Enroll + Top10perc + Top25perc + 
##     F.Undergrad + P.Undergrad + Outstate + Room.Board + Books + 
##     Personal + PhD + Terminal + perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - Books        1      91.7 123423 4044.3
## - Top10perc    1      93.4 123425 4044.3
## - Accept       1     109.1 123441 4044.4
## - Enroll       1     141.2 123473 4044.6
## - F.Undergrad  1     216.4 123548 4045.1
## - Terminal     1     237.6 123569 4045.2
## - PhD          1     400.2 123732 4046.2
## - Private      1     613.0 123945 4047.5
## - Personal     1     788.4 124120 4048.6
## - Top25perc    1     978.6 124310 4049.8
## <none>                     123332 4050.4
## - Apps         1    1426.8 124759 4052.6
## - Room.Board   1    1705.0 125037 4054.4
## - Expend       1    1854.2 125186 4055.3
## - P.Undergrad  1    2356.0 125688 4058.4
## - Outstate     1    3055.0 126387 4062.7
## - perc.alumni  1    5207.4 128539 4075.8
## 
## Step:  AIC=4044.27
## Grad.Rate ~ Private + Apps + Accept + Enroll + Top10perc + Top25perc + 
##     F.Undergrad + P.Undergrad + Outstate + Room.Board + Personal + 
##     PhD + Terminal + perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - Top10perc    1      86.7 123510 4038.2
## - Accept       1     110.1 123533 4038.3
## - Enroll       1     140.9 123564 4038.5
## - F.Undergrad  1     218.1 123641 4039.0
## - Terminal     1     279.1 123703 4039.4
## - PhD          1     469.5 123893 4040.6
## - Private      1     599.2 124023 4041.4
## - Personal     1     908.6 124332 4043.3
## - Top25perc    1     965.8 124389 4043.7
## <none>                     123423 4044.3
## - Apps         1    1425.9 124849 4046.5
## - Room.Board   1    1638.9 125062 4047.9
## - Expend       1    1885.5 125309 4049.4
## - P.Undergrad  1    2365.4 125789 4052.4
## - Outstate     1    3107.2 126531 4056.9
## - perc.alumni  1    5293.4 128717 4070.2
## 
## Step:  AIC=4038.16
## Grad.Rate ~ Private + Apps + Accept + Enroll + Top25perc + F.Undergrad + 
##     P.Undergrad + Outstate + Room.Board + Personal + PhD + Terminal + 
##     perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - Enroll       1     181.1 123691 4032.6
## - Accept       1     208.0 123718 4032.8
## - F.Undergrad  1     227.5 123738 4032.9
## - Terminal     1     315.0 123825 4033.5
## - PhD          1     533.2 124043 4034.9
## - Private      1     628.0 124138 4035.4
## - Personal     1     902.6 124413 4037.2
## <none>                     123510 4038.2
## - Room.Board   1    1602.8 125113 4041.5
## - Expend       1    1816.3 125326 4042.9
## - Apps         1    1835.9 125346 4043.0
## - P.Undergrad  1    2491.0 126001 4047.0
## - Outstate     1    3241.9 126752 4051.6
## - Top25perc    1    4063.5 127574 4056.7
## - perc.alumni  1    5334.0 128844 4064.4
## 
## Step:  AIC=4032.65
## Grad.Rate ~ Private + Apps + Accept + Top25perc + F.Undergrad + 
##     P.Undergrad + Outstate + Room.Board + Personal + PhD + Terminal + 
##     perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - F.Undergrad  1      53.0 123744 4026.3
## - Accept       1      95.1 123786 4026.6
## - Terminal     1     338.0 124029 4028.1
## - PhD          1     545.3 124236 4029.4
## - Private      1     631.2 124322 4029.9
## - Personal     1     895.3 124586 4031.6
## <none>                     123691 4032.6
## - Room.Board   1    1523.1 125214 4035.5
## - Expend       1    1715.1 125406 4036.7
## - Apps         1    1720.1 125411 4036.7
## - P.Undergrad  1    2613.8 126305 4042.2
## - Outstate     1    3190.0 126881 4045.8
## - Top25perc    1    4126.3 127817 4051.5
## - perc.alumni  1    5621.6 129313 4060.5
## 
## Step:  AIC=4026.32
## Grad.Rate ~ Private + Apps + Accept + Top25perc + P.Undergrad + 
##     Outstate + Room.Board + Personal + PhD + Terminal + perc.alumni + 
##     Expend
## 
##               Df Sum of Sq    RSS    AIC
## - Accept       1     251.7 123996 4021.2
## - Terminal     1     350.2 124094 4021.9
## - PhD          1     549.6 124294 4023.1
## - Private      1     749.8 124494 4024.4
## - Personal     1     973.0 124717 4025.8
## <none>                     123744 4026.3
## - Room.Board   1    1572.3 125317 4029.5
## - Expend       1    1750.9 125495 4030.6
## - Apps         1    1775.8 125520 4030.7
## - P.Undergrad  1    3195.5 126940 4039.5
## - Outstate     1    3415.9 127160 4040.8
## - Top25perc    1    4094.9 127839 4045.0
## - perc.alumni  1    5579.5 129324 4053.9
## 
## Step:  AIC=4021.25
## Grad.Rate ~ Private + Apps + Top25perc + P.Undergrad + Outstate + 
##     Room.Board + Personal + PhD + Terminal + perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - Terminal     1     391.0 124387 4017.0
## - PhD          1     524.0 124520 4017.9
## - Private      1     785.5 124781 4019.5
## - Personal     1     992.0 124988 4020.8
## <none>                     123996 4021.2
## - Expend       1    1512.5 125508 4024.0
## - Room.Board   1    1705.6 125701 4025.2
## - Outstate     1    3221.2 127217 4034.5
## - P.Undergrad  1    3449.0 127445 4035.9
## - Top25perc    1    4503.8 128500 4042.3
## - Apps         1    5016.2 129012 4045.4
## - perc.alumni  1    5748.0 129744 4049.8
## 
## Step:  AIC=4017.04
## Grad.Rate ~ Private + Apps + Top25perc + P.Undergrad + Outstate + 
##     Room.Board + Personal + PhD + perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - PhD          1     154.1 124541 4011.3
## - Private      1     953.1 125340 4016.3
## - Personal     1     980.8 125368 4016.5
## <none>                     124387 4017.0
## - Room.Board   1    1537.9 125925 4019.9
## - Expend       1    1543.5 125930 4020.0
## - Outstate     1    3060.7 127448 4029.3
## - P.Undergrad  1    3577.8 127965 4032.4
## - Top25perc    1    4346.4 128733 4037.1
## - Apps         1    5047.6 129434 4041.3
## - perc.alumni  1    5589.7 129977 4044.5
## 
## Step:  AIC=4011.34
## Grad.Rate ~ Private + Apps + Top25perc + P.Undergrad + Outstate + 
##     Room.Board + Personal + perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - Private      1     804.4 125345 4009.7
## - Personal     1     979.0 125520 4010.8
## <none>                     124541 4011.3
## - Expend       1    1480.2 126021 4013.9
## - Room.Board   1    1648.1 126189 4014.9
## - P.Undergrad  1    3464.5 128005 4026.0
## - Outstate     1    3498.1 128039 4026.2
## - Apps         1    5102.6 129644 4035.9
## - Top25perc    1    5446.3 129987 4037.9
## - perc.alumni  1    5756.0 130297 4039.8
## 
## Step:  AIC=4009.69
## Grad.Rate ~ Apps + Top25perc + P.Undergrad + Outstate + Room.Board + 
##     Personal + perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - Personal     1    1013.1 126358 4009.3
## <none>                     125345 4009.7
## - Expend       1    1522.0 126867 4012.4
## - Room.Board   1    1911.9 127257 4014.8
## - Apps         1    4298.5 129644 4029.2
## - P.Undergrad  1    4322.9 129668 4029.4
## - Top25perc    1    5168.7 130514 4034.4
## - Outstate     1    5492.6 130838 4036.4
## - perc.alumni  1    6198.0 131543 4040.5
## 
## Step:  AIC=4009.29
## Grad.Rate ~ Apps + Top25perc + P.Undergrad + Outstate + Room.Board + 
##     perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## <none>                     126358 4009.3
## - Expend       1    1814.5 128173 4013.7
## - Room.Board   1    2158.0 128516 4015.8
## - Apps         1    4083.7 130442 4027.3
## - Top25perc    1    5047.0 131405 4033.1
## - P.Undergrad  1    5385.1 131743 4035.1
## - Outstate     1    6221.0 132579 4040.0
## - perc.alumni  1    6952.7 133311 4044.3
summary(step_back_BIC)
## 
## Call:
## lm(formula = Grad.Rate ~ Apps + Top25perc + P.Undergrad + Outstate + 
##     Room.Board + perc.alumni + Expend, data = College)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -53.627  -7.564  -0.916   7.280  54.100 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 30.3244557  2.1568667  14.059  < 2e-16 ***
## Apps         0.0007374  0.0001479   4.985 7.65e-07 ***
## Top25perc    0.1692769  0.0305437   5.542 4.11e-08 ***
## P.Undergrad -0.0019991  0.0003492  -5.725 1.48e-08 ***
## Outstate     0.0012635  0.0002054   6.153 1.22e-09 ***
## Room.Board   0.0020719  0.0005717   3.624 0.000309 ***
## perc.alumni  0.3123681  0.0480208   6.505 1.40e-10 ***
## Expend      -0.0004280  0.0001288  -3.323 0.000932 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 12.82 on 769 degrees of freedom
## Multiple R-squared:  0.4482, Adjusted R-squared:  0.4431 
## F-statistic: 89.22 on 7 and 769 DF,  p-value: < 2.2e-16
### Método forward. Para o método forward devemos definir o escopo da seleção
### (menor e maior modelo). O menor seria o modelo nulo (apenas com o intercepto),
### enquanto o maior seria o modelo com todas as covariáveis.
aj_lower <- lm(Grad.Rate~1, data = College)

aj_upper <- lm(Grad.Rate~., data = College)
formula(aj_upper)
## Grad.Rate ~ Private + Apps + Accept + Enroll + Top10perc + Top25perc + 
##     F.Undergrad + P.Undergrad + Outstate + Room.Board + Books + 
##     Personal + PhD + Terminal + S.F.Ratio + perc.alumni + Expend
step_for_BIC <- step(aj_lower, direction = "forward", scope=formula(aj_upper), 
                     data = College, k = log(nrow(College)))
## Start:  AIC=4424.63
## Grad.Rate ~ 1
## 
##               Df Sum of Sq    RSS    AIC
## + Outstate     1     74732 154245 4124.3
## + Top10perc    1     56103 172875 4212.9
## + perc.alumni  1     55179 173798 4217.0
## + Top25perc    1     52160 176817 4230.4
## + Room.Board   1     41348 187630 4276.5
## + Expend       1     34889 194089 4302.8
## + Private      1     25876 203102 4338.1
## + S.F.Ratio    1     21540 207437 4354.5
## + PhD          1     21306 207671 4355.4
## + Terminal     1     19194 209783 4363.3
## + Personal     1     16611 212366 4372.8
## + P.Undergrad  1     15124 213853 4378.2
## + Apps         1      4931 224046 4414.4
## <none>                     228977 4424.6
## + F.Undergrad  1      1421 227556 4426.4
## + Accept       1      1037 227940 4427.8
## + Enroll       1       114 228863 4430.9
## + Books        1         0 228977 4431.3
## 
## Step:  AIC=4124.31
## Grad.Rate ~ Outstate
## 
##               Df Sum of Sq    RSS    AIC
## + Top25perc    1   11767.7 142478 4069.3
## + Top10perc    1   10107.6 144138 4078.3
## + perc.alumni  1    9444.9 144800 4081.9
## + Apps         1    3201.7 151044 4114.7
## + P.Undergrad  1    3079.0 151166 4115.3
## + Personal     1    2438.8 151807 4118.6
## + PhD          1    1995.9 152250 4120.8
## + Accept       1    1541.6 152704 4123.2
## <none>                     154245 4124.3
## + Room.Board   1    1048.3 153197 4125.7
## + Enroll       1    1037.1 153208 4125.7
## + Terminal     1     875.4 153370 4126.5
## + F.Undergrad  1     475.1 153770 4128.6
## + Private      1     137.7 154108 4130.3
## + Books        1     102.5 154143 4130.4
## + S.F.Ratio    1      34.8 154211 4130.8
## + Expend       1      15.0 154230 4130.9
## 
## Step:  AIC=4069.3
## Grad.Rate ~ Outstate + Top25perc
## 
##               Df Sum of Sq    RSS    AIC
## + perc.alumni  1    5997.5 136480 4042.5
## + P.Undergrad  1    4196.4 138281 4052.7
## + Personal     1    3376.9 139101 4057.3
## + Private      1    1517.5 140960 4067.6
## <none>                     142478 4069.3
## + Expend       1     957.6 141520 4070.7
## + Room.Board   1     931.7 141546 4070.9
## + Books        1     496.1 141982 4073.2
## + F.Undergrad  1     334.0 142144 4074.1
## + Apps         1     292.3 142186 4074.4
## + Terminal     1     259.5 142218 4074.5
## + Top10perc    1     223.9 142254 4074.7
## + S.F.Ratio    1      87.5 142390 4075.5
## + Accept       1      52.3 142425 4075.7
## + Enroll       1      40.1 142438 4075.7
## + PhD          1      16.1 142462 4075.9
## 
## Step:  AIC=4042.54
## Grad.Rate ~ Outstate + Top25perc + perc.alumni
## 
##               Df Sum of Sq    RSS    AIC
## + P.Undergrad  1   2580.25 133900 4034.4
## + Personal     1   2096.79 134383 4037.2
## + Room.Board   1   1919.65 134561 4038.2
## + Apps         1   1347.54 135133 4041.5
## <none>                     136480 4042.5
## + Expend       1    958.10 135522 4043.7
## + Accept       1    758.79 135721 4044.9
## + Private      1    555.35 135925 4046.0
## + S.F.Ratio    1    370.29 136110 4047.1
## + Books        1    213.78 136266 4048.0
## + Terminal     1    178.28 136302 4048.2
## + Top10perc    1     97.21 136383 4048.6
## + Enroll       1     85.88 136394 4048.7
## + F.Undergrad  1      1.19 136479 4049.2
## + PhD          1      0.00 136480 4049.2
## 
## Step:  AIC=4034.37
## Grad.Rate ~ Outstate + Top25perc + perc.alumni + P.Undergrad
## 
##               Df Sum of Sq    RSS    AIC
## + Apps         1    3864.5 130035 4018.3
## + Accept       1    2892.9 131007 4024.1
## + Room.Board   1    2493.1 131407 4026.4
## + Enroll       1    1470.5 132430 4032.4
## + Personal     1    1213.4 132687 4033.9
## <none>                     133900 4034.4
## + F.Undergrad  1     953.7 132946 4035.5
## + Expend       1     668.5 133231 4037.1
## + S.F.Ratio    1     586.1 133314 4037.6
## + PhD          1     189.5 133711 4039.9
## + Books        1     125.1 133775 4040.3
## + Top10perc    1      64.3 133836 4040.6
## + Private      1      35.8 133864 4040.8
## + Terminal     1       0.0 133900 4041.0
## 
## Step:  AIC=4018.27
## Grad.Rate ~ Outstate + Top25perc + perc.alumni + P.Undergrad + 
##     Apps
## 
##               Df Sum of Sq    RSS    AIC
## + Room.Board   1   1862.58 128173 4013.7
## + Personal     1   1533.45 128502 4015.7
## + Expend       1   1519.13 128516 4015.8
## + Private      1   1163.67 128872 4017.9
## <none>                     130035 4018.3
## + F.Undergrad  1    736.15 129299 4020.5
## + Enroll       1    389.66 129646 4022.6
## + S.F.Ratio    1    282.41 129753 4023.2
## + Books        1    210.11 129825 4023.7
## + Terminal     1    134.41 129901 4024.1
## + Accept       1    107.62 129928 4024.3
## + Top10perc    1      4.46 130031 4024.9
## + PhD          1      2.19 130033 4024.9
## 
## Step:  AIC=4013.71
## Grad.Rate ~ Outstate + Top25perc + perc.alumni + P.Undergrad + 
##     Apps + Room.Board
## 
##               Df Sum of Sq    RSS    AIC
## + Expend       1   1814.50 126358 4009.3
## + Personal     1   1305.58 126867 4012.4
## <none>                     128173 4013.7
## + Private      1    890.33 127283 4015.0
## + F.Undergrad  1    465.62 127707 4017.5
## + Books        1    372.34 127801 4018.1
## + S.F.Ratio    1    353.01 127820 4018.2
## + Terminal     1    280.14 127893 4018.7
## + Enroll       1    165.70 128007 4019.4
## + Accept       1     34.78 128138 4020.2
## + PhD          1      1.56 128171 4020.4
## + Top10perc    1      1.43 128171 4020.4
## 
## Step:  AIC=4009.29
## Grad.Rate ~ Outstate + Top25perc + perc.alumni + P.Undergrad + 
##     Apps + Room.Board + Expend
## 
##               Df Sum of Sq    RSS    AIC
## <none>                     126358 4009.3
## + Personal     1   1013.08 125345 4009.7
## + Private      1    838.46 125520 4010.8
## + F.Undergrad  1    639.71 125719 4012.0
## + Accept       1    348.76 126010 4013.8
## + Top10perc    1    289.00 126069 4014.2
## + Books        1    275.25 126083 4014.3
## + Enroll       1    271.29 126087 4014.3
## + Terminal     1    189.86 126169 4014.8
## + PhD          1      4.21 126354 4015.9
## + S.F.Ratio    1      3.90 126354 4015.9
summary(step_for_BIC)
## 
## Call:
## lm(formula = Grad.Rate ~ Outstate + Top25perc + perc.alumni + 
##     P.Undergrad + Apps + Room.Board + Expend, data = College)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -53.627  -7.564  -0.916   7.280  54.100 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 30.3244557  2.1568667  14.059  < 2e-16 ***
## Outstate     0.0012635  0.0002054   6.153 1.22e-09 ***
## Top25perc    0.1692769  0.0305437   5.542 4.11e-08 ***
## perc.alumni  0.3123681  0.0480208   6.505 1.40e-10 ***
## P.Undergrad -0.0019991  0.0003492  -5.725 1.48e-08 ***
## Apps         0.0007374  0.0001479   4.985 7.65e-07 ***
## Room.Board   0.0020719  0.0005717   3.624 0.000309 ***
## Expend      -0.0004280  0.0001288  -3.323 0.000932 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 12.82 on 769 degrees of freedom
## Multiple R-squared:  0.4482, Adjusted R-squared:  0.4431 
## F-statistic: 89.22 on 7 and 769 DF,  p-value: < 2.2e-16
### Finalmente, o algoritmo que considera tanto exclusão quanto inclusão de
### covariáveis a cada passo
step_both_BIC <- step(aj_full, direction = "both", data = College, k = log(nrow(College)))
## Start:  AIC=4056.77
## Grad.Rate ~ Private + Apps + Accept + Enroll + Top10perc + Top25perc + 
##     F.Undergrad + P.Undergrad + Outstate + Room.Board + Books + 
##     Personal + PhD + Terminal + S.F.Ratio + perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - S.F.Ratio    1      36.8 123332 4050.4
## - Books        1      94.1 123389 4050.7
## - Top10perc    1      95.1 123390 4050.7
## - Accept       1     105.8 123401 4050.8
## - Enroll       1     142.2 123437 4051.0
## - F.Undergrad  1     224.5 123519 4051.5
## - Terminal     1     233.4 123528 4051.6
## - PhD          1     383.4 123678 4052.5
## - Private      1     645.3 123940 4054.2
## - Personal     1     758.7 124054 4054.9
## - Top25perc    1     981.7 124277 4056.3
## <none>                     123295 4056.8
## - Apps         1    1403.4 124698 4058.9
## - Expend       1    1424.2 124719 4059.0
## - Room.Board   1    1705.5 125000 4060.8
## - P.Undergrad  1    2348.7 125644 4064.8
## - Outstate     1    3086.3 126381 4069.3
## - perc.alumni  1    5241.6 128537 4082.5
## 
## Step:  AIC=4050.35
## Grad.Rate ~ Private + Apps + Accept + Enroll + Top10perc + Top25perc + 
##     F.Undergrad + P.Undergrad + Outstate + Room.Board + Books + 
##     Personal + PhD + Terminal + perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - Books        1      91.7 123423 4044.3
## - Top10perc    1      93.4 123425 4044.3
## - Accept       1     109.1 123441 4044.4
## - Enroll       1     141.2 123473 4044.6
## - F.Undergrad  1     216.4 123548 4045.1
## - Terminal     1     237.6 123569 4045.2
## - PhD          1     400.2 123732 4046.2
## - Private      1     613.0 123945 4047.5
## - Personal     1     788.4 124120 4048.6
## - Top25perc    1     978.6 124310 4049.8
## <none>                     123332 4050.4
## - Apps         1    1426.8 124759 4052.6
## - Room.Board   1    1705.0 125037 4054.4
## - Expend       1    1854.2 125186 4055.3
## + S.F.Ratio    1      36.8 123295 4056.8
## - P.Undergrad  1    2356.0 125688 4058.4
## - Outstate     1    3055.0 126387 4062.7
## - perc.alumni  1    5207.4 128539 4075.8
## 
## Step:  AIC=4044.27
## Grad.Rate ~ Private + Apps + Accept + Enroll + Top10perc + Top25perc + 
##     F.Undergrad + P.Undergrad + Outstate + Room.Board + Personal + 
##     PhD + Terminal + perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - Top10perc    1      86.7 123510 4038.2
## - Accept       1     110.1 123533 4038.3
## - Enroll       1     140.9 123564 4038.5
## - F.Undergrad  1     218.1 123641 4039.0
## - Terminal     1     279.1 123703 4039.4
## - PhD          1     469.5 123893 4040.6
## - Private      1     599.2 124023 4041.4
## - Personal     1     908.6 124332 4043.3
## - Top25perc    1     965.8 124389 4043.7
## <none>                     123423 4044.3
## - Apps         1    1425.9 124849 4046.5
## - Room.Board   1    1638.9 125062 4047.9
## - Expend       1    1885.5 125309 4049.4
## + Books        1      91.7 123332 4050.4
## + S.F.Ratio    1      34.3 123389 4050.7
## - P.Undergrad  1    2365.4 125789 4052.4
## - Outstate     1    3107.2 126531 4056.9
## - perc.alumni  1    5293.4 128717 4070.2
## 
## Step:  AIC=4038.16
## Grad.Rate ~ Private + Apps + Accept + Enroll + Top25perc + F.Undergrad + 
##     P.Undergrad + Outstate + Room.Board + Personal + PhD + Terminal + 
##     perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - Enroll       1     181.1 123691 4032.6
## - Accept       1     208.0 123718 4032.8
## - F.Undergrad  1     227.5 123738 4032.9
## - Terminal     1     315.0 123825 4033.5
## - PhD          1     533.2 124043 4034.9
## - Private      1     628.0 124138 4035.4
## - Personal     1     902.6 124413 4037.2
## <none>                     123510 4038.2
## - Room.Board   1    1602.8 125113 4041.5
## - Expend       1    1816.3 125326 4042.9
## - Apps         1    1835.9 125346 4043.0
## + Top10perc    1      86.7 123423 4044.3
## + Books        1      85.0 123425 4044.3
## + S.F.Ratio    1      32.9 123477 4044.6
## - P.Undergrad  1    2491.0 126001 4047.0
## - Outstate     1    3241.9 126752 4051.6
## - Top25perc    1    4063.5 127574 4056.7
## - perc.alumni  1    5334.0 128844 4064.4
## 
## Step:  AIC=4032.65
## Grad.Rate ~ Private + Apps + Accept + Top25perc + F.Undergrad + 
##     P.Undergrad + Outstate + Room.Board + Personal + PhD + Terminal + 
##     perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - F.Undergrad  1      53.0 123744 4026.3
## - Accept       1      95.1 123786 4026.6
## - Terminal     1     338.0 124029 4028.1
## - PhD          1     545.3 124236 4029.4
## - Private      1     631.2 124322 4029.9
## - Personal     1     895.3 124586 4031.6
## <none>                     123691 4032.6
## - Room.Board   1    1523.1 125214 4035.5
## - Expend       1    1715.1 125406 4036.7
## - Apps         1    1720.1 125411 4036.7
## + Enroll       1     181.1 123510 4038.2
## + Top10perc    1     126.9 123564 4038.5
## + Books        1      83.2 123608 4038.8
## + S.F.Ratio    1      31.5 123660 4039.1
## - P.Undergrad  1    2613.8 126305 4042.2
## - Outstate     1    3190.0 126881 4045.8
## - Top25perc    1    4126.3 127817 4051.5
## - perc.alumni  1    5621.6 129313 4060.5
## 
## Step:  AIC=4026.32
## Grad.Rate ~ Private + Apps + Accept + Top25perc + P.Undergrad + 
##     Outstate + Room.Board + Personal + PhD + Terminal + perc.alumni + 
##     Expend
## 
##               Df Sum of Sq    RSS    AIC
## - Accept       1     251.7 123996 4021.2
## - Terminal     1     350.2 124094 4021.9
## - PhD          1     549.6 124294 4023.1
## - Private      1     749.8 124494 4024.4
## - Personal     1     973.0 124717 4025.8
## <none>                     123744 4026.3
## - Room.Board   1    1572.3 125317 4029.5
## - Expend       1    1750.9 125495 4030.6
## - Apps         1    1775.8 125520 4030.7
## + Top10perc    1     102.4 123642 4032.3
## + Books        1      85.4 123659 4032.4
## + F.Undergrad  1      53.0 123691 4032.6
## + S.F.Ratio    1      26.2 123718 4032.8
## + Enroll       1       6.5 123738 4032.9
## - P.Undergrad  1    3195.5 126940 4039.5
## - Outstate     1    3415.9 127160 4040.8
## - Top25perc    1    4094.9 127839 4045.0
## - perc.alumni  1    5579.5 129324 4053.9
## 
## Step:  AIC=4021.25
## Grad.Rate ~ Private + Apps + Top25perc + P.Undergrad + Outstate + 
##     Room.Board + Personal + PhD + Terminal + perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - Terminal     1     391.0 124387 4017.0
## - PhD          1     524.0 124520 4017.9
## - Private      1     785.5 124781 4019.5
## - Personal     1     992.0 124988 4020.8
## <none>                     123996 4021.2
## - Expend       1    1512.5 125508 4024.0
## - Room.Board   1    1705.6 125701 4025.2
## + Accept       1     251.7 123744 4026.3
## + F.Undergrad  1     209.5 123786 4026.6
## + Top10perc    1     196.1 123800 4026.7
## + Books        1      86.5 123909 4027.4
## + Enroll       1      63.4 123932 4027.5
## + S.F.Ratio    1      25.0 123971 4027.7
## - Outstate     1    3221.2 127217 4034.5
## - P.Undergrad  1    3449.0 127445 4035.9
## - Top25perc    1    4503.8 128500 4042.3
## - Apps         1    5016.2 129012 4045.4
## - perc.alumni  1    5748.0 129744 4049.8
## 
## Step:  AIC=4017.04
## Grad.Rate ~ Private + Apps + Top25perc + P.Undergrad + Outstate + 
##     Room.Board + Personal + PhD + perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - PhD          1     154.1 124541 4011.3
## - Private      1     953.1 125340 4016.3
## - Personal     1     980.8 125368 4016.5
## <none>                     124387 4017.0
## - Room.Board   1    1537.9 125925 4019.9
## - Expend       1    1543.5 125930 4020.0
## + Terminal     1     391.0 123996 4021.2
## + Accept       1     292.4 124094 4021.9
## + Top10perc    1     261.5 124125 4022.1
## + F.Undergrad  1     250.4 124136 4022.1
## + Books        1     134.6 124252 4022.9
## + Enroll       1      78.0 124309 4023.2
## + S.F.Ratio    1      27.9 124359 4023.5
## - Outstate     1    3060.7 127448 4029.3
## - P.Undergrad  1    3577.8 127965 4032.4
## - Top25perc    1    4346.4 128733 4037.1
## - Apps         1    5047.6 129434 4041.3
## - perc.alumni  1    5589.7 129977 4044.5
## 
## Step:  AIC=4011.34
## Grad.Rate ~ Private + Apps + Top25perc + P.Undergrad + Outstate + 
##     Room.Board + Personal + perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - Private      1     804.4 125345 4009.7
## - Personal     1     979.0 125520 4010.8
## <none>                     124541 4011.3
## - Expend       1    1480.2 126021 4013.9
## - Room.Board   1    1648.1 126189 4014.9
## + Top10perc    1     280.4 124261 4016.2
## + Accept       1     240.4 124301 4016.5
## + F.Undergrad  1     216.4 124325 4016.6
## + Books        1     159.1 124382 4017.0
## + PhD          1     154.1 124387 4017.0
## + Enroll       1      60.7 124480 4017.6
## + S.F.Ratio    1      37.7 124503 4017.8
## + Terminal     1      21.1 124520 4017.9
## - P.Undergrad  1    3464.5 128005 4026.0
## - Outstate     1    3498.1 128039 4026.2
## - Apps         1    5102.6 129644 4035.9
## - Top25perc    1    5446.3 129987 4037.9
## - perc.alumni  1    5756.0 130297 4039.8
## 
## Step:  AIC=4009.69
## Grad.Rate ~ Apps + Top25perc + P.Undergrad + Outstate + Room.Board + 
##     Personal + perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## - Personal     1    1013.1 126358 4009.3
## <none>                     125345 4009.7
## + Private      1     804.4 124541 4011.3
## - Expend       1    1522.0 126867 4012.4
## + F.Undergrad  1     461.7 124884 4013.5
## + Accept       1     325.6 125020 4014.3
## + Top10perc    1     317.3 125028 4014.4
## - Room.Board   1    1911.9 127257 4014.8
## + Terminal     1     188.2 125157 4015.2
## + Enroll       1     177.9 125167 4015.2
## + Books        1     128.7 125217 4015.5
## + PhD          1       5.4 125340 4016.3
## + S.F.Ratio    1       0.0 125345 4016.3
## - Apps         1    4298.5 129644 4029.2
## - P.Undergrad  1    4322.9 129668 4029.4
## - Top25perc    1    5168.7 130514 4034.4
## - Outstate     1    5492.6 130838 4036.4
## - perc.alumni  1    6198.0 131543 4040.5
## 
## Step:  AIC=4009.29
## Grad.Rate ~ Apps + Top25perc + P.Undergrad + Outstate + Room.Board + 
##     perc.alumni + Expend
## 
##               Df Sum of Sq    RSS    AIC
## <none>                     126358 4009.3
## + Personal     1    1013.1 125345 4009.7
## + Private      1     838.5 125520 4010.8
## + F.Undergrad  1     639.7 125719 4012.0
## - Expend       1    1814.5 128173 4013.7
## + Accept       1     348.8 126010 4013.8
## + Top10perc    1     289.0 126069 4014.2
## + Books        1     275.2 126083 4014.3
## + Enroll       1     271.3 126087 4014.3
## + Terminal     1     189.9 126169 4014.8
## - Room.Board   1    2158.0 128516 4015.8
## + PhD          1       4.2 126354 4015.9
## + S.F.Ratio    1       3.9 126354 4015.9
## - Apps         1    4083.7 130442 4027.3
## - Top25perc    1    5047.0 131405 4033.1
## - P.Undergrad  1    5385.1 131743 4035.1
## - Outstate     1    6221.0 132579 4040.0
## - perc.alumni  1    6952.7 133311 4044.3
summary(step_both_BIC)
## 
## Call:
## lm(formula = Grad.Rate ~ Apps + Top25perc + P.Undergrad + Outstate + 
##     Room.Board + perc.alumni + Expend, data = College)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -53.627  -7.564  -0.916   7.280  54.100 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 30.3244557  2.1568667  14.059  < 2e-16 ***
## Apps         0.0007374  0.0001479   4.985 7.65e-07 ***
## Top25perc    0.1692769  0.0305437   5.542 4.11e-08 ***
## P.Undergrad -0.0019991  0.0003492  -5.725 1.48e-08 ***
## Outstate     0.0012635  0.0002054   6.153 1.22e-09 ***
## Room.Board   0.0020719  0.0005717   3.624 0.000309 ***
## perc.alumni  0.3123681  0.0480208   6.505 1.40e-10 ***
## Expend      -0.0004280  0.0001288  -3.323 0.000932 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 12.82 on 769 degrees of freedom
## Multiple R-squared:  0.4482, Adjusted R-squared:  0.4431 
## F-statistic: 89.22 on 7 and 769 DF,  p-value: < 2.2e-16
### Vamos comparar os ajustes.
data.frame(compareCoefs(step_back_BIC, step_for_BIC, step_both_BIC))
## Calls:
## 1: lm(formula = Grad.Rate ~ Apps + Top25perc + P.Undergrad + Outstate + 
##   Room.Board + perc.alumni + Expend, data = College)
## 2: lm(formula = Grad.Rate ~ Outstate + Top25perc + perc.alumni + 
##   P.Undergrad + Apps + Room.Board + Expend, data = College)
## 3: lm(formula = Grad.Rate ~ Apps + Top25perc + P.Undergrad + Outstate + 
##   Room.Board + perc.alumni + Expend, data = College)
## 
##               Model 1   Model 2   Model 3
## (Intercept)     30.32     30.32     30.32
## SE               2.16      2.16      2.16
##                                          
## Apps         0.000737  0.000737  0.000737
## SE           0.000148  0.000148  0.000148
##                                          
## Top25perc      0.1693    0.1693    0.1693
## SE             0.0305    0.0305    0.0305
##                                          
## P.Undergrad -0.001999 -0.001999 -0.001999
## SE           0.000349  0.000349  0.000349
##                                          
## Outstate     0.001264  0.001264  0.001264
## SE           0.000205  0.000205  0.000205
##                                          
## Room.Board   0.002072  0.002072  0.002072
## SE           0.000572  0.000572  0.000572
##                                          
## perc.alumni     0.312     0.312     0.312
## SE              0.048     0.048     0.048
##                                          
## Expend      -0.000428 -0.000428 -0.000428
## SE           0.000129  0.000129  0.000129
## 
##                    Model.1       Model.2       Model.3
## X.Intercept. 30.3244557458 30.3244557458 30.3244557458
## SE            2.1568666696  2.1568666696  2.1568666696
## X                       NA            NA            NA
## Apps          0.0007374001  0.0007374001  0.0007374001
## SE.1          0.0001479164  0.0001479164  0.0001479164
## X.1                     NA            NA            NA
## Top25perc     0.1692769245  0.1692769245  0.1692769245
## SE.2          0.0305437176  0.0305437176  0.0305437176
## X.2                     NA            NA            NA
## P.Undergrad  -0.0019991418 -0.0019991418 -0.0019991418
## SE.3          0.0003492104  0.0003492104  0.0003492104
## X.3                     NA            NA            NA
## Outstate      0.0012635375  0.0012635375  0.0012635375
## SE.4          0.0002053515  0.0002053515  0.0002053515
## X.4                     NA            NA            NA
## Room.Board    0.0020719407  0.0020719407  0.0020719407
## SE.5          0.0005717359  0.0005717359  0.0005717359
## X.5                     NA            NA            NA
## perc.alumni   0.3123680898  0.3123680898  0.3123680898
## SE.6          0.0480208208  0.0480208208  0.0480208208
## X.6                     NA            NA            NA
## Expend       -0.0004280405 -0.0004280405 -0.0004280405
## SE.7          0.0001288088  0.0001288088  0.0001288088
## X.7                     NA            NA            NA

```